On Eigenvectors for Semisimple Elements in
Actions of Algebraic Groups
by
Darren John Kenneally
Jesus College
Cambridge
A dissertation submitted for the degree
of Doctor of Philosophy to the
University of Cambridge
Abstract
Let G be a simple simply connected algebraic group defined over an algebraically closed
field K and V an irreducible module defined over K on which G acts. Let E denote
the set of vectors in V which are eigenvectors for some non-central semisimple element
of G and some eigenvalue in K∗. We prove, with a short list of possible exceptions, that
the dimension of E is strictly less than the dimension of V provided dimV > dimG + 2
and that there is equality otherwise. In particular, by considering only the eigenvalue
1, it follows that the closure of the union of fixed point spaces of non-central semisimple
elements has dimension strictly less than the dimension of V provided dimV > dimG+2,
with a short list of possible exceptions.
In the majority of cases we consider modules for which dimV > dimG + 2 where we
perform an analysis of weights. In many of these cases we prove that, for any non-central
semisimple element and any eigenvalue, the codimension of the eigenspace exceeds dimG.
In more difficult cases, when dimV is only slightly larger than dimG + 2, we subdivide
the analysis according to the type of the centraliser of the semisimple element. Here we
prove for each type a slightly weaker inequality which still suffices to establish the main
result. Finally, for the relatively few modules satisfying dimV 6 dimG+2, an immediate
observation yields the result for dimV < dimB where B is a Borel subgroup of G, while
in other cases we argue directly.
i
Preface
Declaration
I hereby declare that this dissertation is not substantially the same as any that I have
submitted for a degree, diploma or other qualification at any other university, and that no
part of it has been or is currently being submitted for any such degree, diploma or other
qualification. I further declare that this dissertation is the result of my own work and
contains nothing which is the outcome of work done in collaboration with others, except
as specified below.
Chapters 3 and 4 are joint work with Dr. Ross Lawther, and Section 3.6 is solely due to
Lawther. All other content is my own work.
ii
Acknowledgements
First I would like to thank my supervisor Dr. Ross Lawther for his guidance, generous
support, encouragement and patience, and for elucidating some beautiful mathematics.
I am indebted to Prof. Jan Saxl for bringing me to Cambridge in the first place and thank
him for his interest in my work.
I have had a wonderful time as a graduate student at Jesus College. I would like to thank
all of my supervisees and, indeed, Dr. Stephen Siklos for trusting me to supervise them.
I owe much to many people at the University of Warwick, where I was lucky to attend
excellent lectures during my undergraduate degree. I would like to thank Prof. Roger
Carter in particular for MA5N9: Representation Theory of the General Linear Group.
I would like to thank Professors Frank Lu¨beck and Alex Zalesski for being generous with
their time to answer all of my queries.
I would like to thank my family for their help throughout my life, in particular my mother.
I thank Jasminder Ghuman for her love, kindness, patience and for giving me confidence
when I found it lacking myself.
I would like to thank my office-mate Alex Frolkin for his advice and for listening as I
gave my views on the world. Also, thanks to Chris Bowman and Matthew Clarke for
their good humour. I have been fortunate to be able to spend time in the company of
my friends Dr. Ben Fairbairn and Dr. Olof Sisask; trips to Chicago, Northern Italy and
F.Y.R. Macedonia spring to mind. I must thank David Stewart for a most worthwhile
hour learning about tensor indecomposable modules. My flat-mates at both Warwick and
Cambridge have kept me sane and often done my washing-up. I would like to thank Mark
Dunning, Nick Jones and Fabio Viola in particular.
I gratefully acknowledge the funding provided by E.P.S.R.C.
This thesis is dedicated to Dr. John Felton.
iii
Contents
1 Introduction 1
2 Preliminaries and statement of main theorem 4
2.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Theorems of Lang and Steinberg . . . . . . . . . . . . . . . . . . . 7
2.2 Premet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Zalesski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Centralisers of semisimple elements . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Lie algebras and a result on nilpotent orbits . . . . . . . . . . . . . . . . . 14
2.6 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Small modules 17
3.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Twisted modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Finite orbit modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Two specific cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Type A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 The action of SL7(K) on L(ω3) . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Techniques for large modules 34
4.1 Analysis of centraliser types . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Adjacency principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Criteria for modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Groups of type An 44
5.1 Initial survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
iv
5.2 Weight string analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Centraliser analysis for L(ω4) and L(ω5) . . . . . . . . . . . . . . . . . . . 60
5.4 Centraliser analysis for L(ω3) . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.5 Groups of low rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5.1 Case I: n = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.2 Case II: n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5.3 Case III: n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5.4 Case IV: n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 Groups of type Bn 100
6.1 Initial survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Even characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Weight string analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Groups of type Cn 123
7.1 Initial survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Even characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Weight string analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8 Groups of type Dn 139
8.1 Initial survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.2 Weight string analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9 Groups of exceptional type 151
9.1 Types E6, E7 and E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2 Type F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.3 Type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10 Twisted modules 158
10.1 Initial survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.2 Weight net analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
11 Concluding remarks 170
A Cliques for (G, V ) = (A12, L(ω3)) 173
v
Chapter 1
Introduction
A very powerful way of approaching groups is through their actions. Actions give an
insight into the structure of a group by representing it as a group of matrices. We shall be
concerned with actions of algebraic groups on vector spaces, both defined over the same
algebraically closed field. We study the building blocks of the theory, namely, irreducible
actions of simple algebraic groups. We assume that simple algebraic groups are simply
connected since this version of the group acts on all of its modules, whereas versions in
other isogeny classes do not.
Let G be a simple simply connected algebraic group defined over an algebraically
closed field K of characteristic p > 0 and let V be an irreducible G-module also defined
over K. In representation theory, important invariants in actions for each group element
are eigenvectors and eigenvalues. The eigenspaces with eigenvalue 1 are of interest since
they are the fixed point spaces CV (g) = {v ∈ V | gv = v} for each g ∈ G. A recent
survey article of Zalesski [29] discusses a number of problems on eigenvalues of elements
in representations of algebraic groups and finite Chevalley groups. He points out that the
usual method of studying eigenspaces is through the theory of weights of representations
of algebraic groups, particularly in relation to semisimple elements.
In this thesis, we consider the question of how likely it is that a vector chosen at random
from V will be an eigenvector for some semisimple element of G. We quickly see that we
should restrict our attention to non-central semisimple elements since central elements
1
act as scalars. We consider the set E of vectors in V which are eigenvectors for some
non-central semisimple element of G and some eigenvalue in K∗. Our aim is to precisely
determine when the dimension of its closure is strictly less than dimV . We show, for all
but a short list of possible exceptions, that there is a clear dichotomy: the dimension of E
is strictly less than V whenever dimV > dimG+ 2, otherwise we have equality. In order
to establish this result we show in most cases that the codimension of the eigenspace of
any non-central semisimple element and any eigenvalue exceeds dimG. In the remaining
cases we require an analysis of the centraliser types of semisimple elements.
Fix a non-central semisimple element s ∈ G and an eigenvalue γ ∈ K∗. We shall
explain the principal method used to calculate a lower bound for the codimension of the
corresponding eigenspace Vγ(s) = {v ∈ V | sv = γv}. Since s is non-central, we deduce
from the structure theory of centralisers of semisimple elements that there is a root α in
the root system of G such that α(s) 6= 1. Each irreducible G-module can be decomposed
as a direct sum of weight spaces. Each weight appears in a weight string for the root α
and two weights are adjacent in such a string if they differ by α. Thus the weight spaces
corresponding to two adjacent weights in a weight string for α cannot both lie in the
eigenspace. We shall introduce this later as the adjacency principle in Section 4.2.
As an application of our main result we make a contribution to the study of regular
orbits. By restricting our main result above to 1-eigenspaces we deduce that the set of
vectors whose stabilisers contain no non-central semisimple element is dense inG whenever
dimV > dimG+ 2 except for some possible exceptions. Thus if it can be shown that at
least one of these vectors has a stabiliser containing no non-trivial unipotent element, it
would follow that such a vector has stabiliser contained in Z(G). Such a vector would lie
in a regular orbit when we pass from the action of G to that of G/M , where M 6 Z(G)
is the kernel of the action of G on V .
One of the motivations for considering regular orbits is the work on base size in the
theory of permutation groups by Burness, Liebeck, Shalev, Guralnick, Saxl, et al. Let G
be a primitive permutation group on a set Ω. We call a subset B ⊂ Ω a base for G if the
pointwise stabiliser of B in G is trivial, i.e.,
⋂
α∈B Gα = 1. Denote by b(G) = b(G,Ω) the
2
minimal size of a base of G. By the O’Nan-Scott theorem [17] G is either almost simple,
affine or lies in one of four other classes. If G is affine then G 6 AGL(V ) where V is a
finite-dimensional vector space of order pn with p prime and n ∈ N. If we interpret V
as the group of translations we may write G = V ⋊ H, where H = G0 is an irreducible
subgroup of GL(V ) and V ∼= (Cp)
n is an elementary abelian regular normal subgroup of
G. Consider the action of H on V and let b(H) denote the minimal size of a base for this
action. We have b(G) = 1 + b(H) and so b(G) = 2 if and only if there is a single vector
in V whose stabiliser in H is trivial, i.e., if and only if H has a regular orbit in its action
on V .
Layout This thesis is organised as follows. In Chapter 2 we assemble the background
results required and state the main theorem of this thesis. The next two chapters reflect
the dichotomy of the main theorem. In Chapter 3 we state and prove results for small
modules of dimension less than or equal to dimG+ 2 and in Chapter 4 we provide tech-
niques for large modules, i.e., modules of dimension exceeding dimG+ 2. In Chapters 5,
6, 7 and 8 we consider the possible p-restricted irreducible modules for groups of types
An, Bn, Cn and Dn respectively. In Chapter 9 we consider such modules for groups of
exceptional type. In Chapter 10 we examine tensor products with twists of p-restricted
irreducible modules for groups of all types thereby completing the investigation. In Chap-
ter 11 we close with a review of the progress we have made and open questions that have
arisen. In Appendix A we provide tables containing sets of weights for a specific action
as detailed in Lemma 5.13 in Chapter 5.
3
Chapter 2
Preliminaries and statement of main
theorem
In this chapter, we define terms and recall key results from the representation theory of
algebraic groups. We shall establish the background to the material addressed in the
chapters to follow. We shall finish the chapter by stating the main theorem of this thesis.
2.1 Representation theory
We refer the reader to [1, Chapter 1], [27], [13] and [19] for full details of the theory below.
Let G be a simple simply connected algebraic group over an algebraically closed field of
characteristic p > 0. Let T be a maximal torus of rank n and B ⊃ T be a Borel subgroup.
Denote by Φ the root system of G with respect to T , and Φ+, Φ− and Π = {α1, . . . , αn} the
positive, negative and simple roots determined by the choice of B. We can analogously
denote the coroot system by Φ∨ and denote by α∨ ∈ Φ∨ the coroot corresponding to
α ∈ Φ. Let ( , ) denote the associated symmetric bilinear form. By setting ‖α‖ = (α, α)
1
2
to be the root length of each α ∈ Φ it can be shown that for each Φ either all roots
have the same length, or there are two root lengths and Φ is partitioned into two sets;
one set denoted ΦL contains long roots and the other denoted ΦS contains short roots.
4
If all roots in Φ are of the same length then we shall regard all roots as short. Each
simple algebraic group has an associated connected Dynkin diagram and we shall use
the standard Bourbaki labelling adopted by Humphreys in [13, p.58]. Let U denote the
unipotent radical of B; there is a semidirect product decomposition B = UT . We denote
by B− the opposite Borel subgroup of G with B ∩B− = T and U− denotes the unipotent
radical of B−. The one-dimensional connected unipotent proper subgroups of U and U−
correspond to elements of Φ and are called root subgroups; these are denoted Xα where
α ∈ Φ and we have U =
∏
α∈Φ+ Xα.
Let X = X(T ) be the character group of G and Y = Y (T ) be the cocharacter group
of G, both with respect to T and let Gm be the one-dimensional multiplicative group
isomorphic to GL1(K). For χ ∈ X and γ ∈ Y since χ ◦ γ ∈ Hom(Gm, Gm) ∼= Z, we
have (χ ◦ γ)(λ) = λn for some n ∈ Z and for all λ ∈ Gm; set 〈χ, γ〉 = n. In this way we
can define a non-degenerate map X × Y → Z by (χ, γ) 7→ 〈χ, γ〉. Let W be the Weyl
group with respect to T . The Weyl group acts transitively on roots of the same length
in Φ; indeed, W acts faithfully on the lattices X and Y . It is generated by elements wα
for α ∈ Φ. It is not too hard to show that W is generated by {wαi}αi∈Π and that it is a
Coxeter group. Each wαi acts on X as follows: wαi(χ) = χ − 〈χ, α
∨
i 〉αi for each χ ∈ X.
Indeed, we have 〈χ, α∨〉 = 2 (χ,α)
(α,α)
.
We shall assume from the outset that all representations of G considered are both
finite-dimensional and rational. Let V be a finite-dimensional G-module over K with
corresponding rational representation ρ : G → GL(V ). Considering V as a T -module,
V is diagonalisable in the sense that V =
⊕
λ∈X Vλ where Vλ = {v ∈ V | ρ(t)v =
λ(t)v for all t ∈ T}. We say that λ is a weight of the representation ρ whenever Vλ 6= 0
and we call Vλ the weight space. The multiplicity of λ in ρ is defined to be dimVλ =: mλ.
Since dimVλ = dimVwλ for all w ∈ W , weights in the same Weyl group orbit have the
same multiplicity. Let {ω1, . . . , ωn} be the basis of ZΦ dual to Π∨ with respect to 〈 , 〉, i.e.,
〈ωi, α
∨
j 〉 = δij for 1 6 i, j 6 n. We call {ω1, . . . , ωn} the fundamental weights and since
G is simply connected, this is a Z-basis of X. We say that a weight λ ∈ X is dominant
if 〈λ, α∨i 〉 > 0 for all i, i.e., λ is a non-negative linear combination of the fundamental
5
weights. We denote the set of dominant weights by X+. From above we can see that W
acts on X via wαi(ωj) = ωj − δijαi and in fact each W -orbit contains a unique dominant
weight. The set of weights of V is a union of W -orbits. We can define a partial ordering
6 on X whereby λ 6 λ′ if and only if λ′ − λ is a non-negative linear combination of
simple roots. A dominant weight λ ∈ X+ is said to be p-restricted if 0 6 〈λ, α∨i 〉 < p for
all i ∈ [1, n]. For convenience we shall use the notation 〈ω, α〉 for 〈ω, α∨〉 in subsequent
chapters.
We shall write each fundamental weight as a linear combination of the simple roots
with coefficients in Q+: see [13, p.69]. In order to save space we shall denote a weight
ω =
∑
aiαi by its coefficients ai, i.e., we shall write ω = a1a2 . . . an.
The following theorem makes it clear that we may study irreducible representations
via their highest weight representations.
Theorem 2.1 (Chevalley). Let G be a semisimple algebraic group over K. Then the
following hold.
(i) Every irreducible rational G-module V has a unique highest weight λ ∈ X+ with re-
spect to the partial ordering on X+. In particular the weight space Vλ has multiplicity
1 and if Vµ 6= 0 then µ 6 λ.
(ii) Two irreducible modules with the same highest weight λ are isomorphic.
(iii) For every λ ∈ X+ there exists an irreducible G-module of highest weight λ.
We may thus denote an irreducible G-module with highest weight λ by L(λ) without
confusion.
2.1.1 Duality
If the G-module L(λ) of highest weight λ is irreducible then the dual module L(λ)∗ =
Hom(L(λ), K) is again an irreducible G-module. It is not too hard to work out that L(λ)∗
has highest weight −w0.λ where w0 ∈ W is the longest word in the Weyl group and is
6
the unique element mapping the positive roots to the negative roots, i.e., w0(Φ
+) = Φ−.
By Chevalley’s theorem we have L(λ)∗ ∼= L(−w0.λ). Now −w0 = id on X when Φ
is of type Bn, Cn, Dn (n even), E7, E8, F4 and G2; in particular, in these cases, an
irreducible G-module is self-dual. In the other cases w0 induces a graph automorphism
of the corresponding Dynkin diagrams. Thus, for example, if the root system is of type
An, the modules L(a1ω1 + · · · + anωn) and L(anω1 + · · · + a1ωn) are dual to each other.
It will suffice to consider one of each dual pair since our calculations will be identical in
both cases.
2.1.2 Theorems of Lang and Steinberg
Steinberg proved the following fundamental dichotomy.
Theorem 2.2 (Steinberg, [26]). Let G be a simple linear algebraic group and σ : G→ G
an endomorphism. Then precisely one of the following holds:
(i) σ is an automorphism of algebraic groups;
(ii) the group Gσ = {g ∈ G | σ(g) = g} of fixed points is finite.
In case (ii) of Theorem 2.2 the endomorphism σ is called a Frobenius map of G. The
group Gσ is a finite group of Lie type.
Given a rational representation ρ : G → GL(V ) and the standard Frobenius map F
relative to p (as defined in [1, §1.17]), we can “twist” by the rth power of F to obtain a
new representation ρ◦F r : G→ GL(V ). We shall write V (r) for the associated G-module.
Theorem 2.3 (Steinberg, [26]). Let G be a simply connected semisimple algebraic group
over Fp and λ ∈ X+. Write λ = λ0 + pλ1 + p2λ2 + . . .+ prλr, the p-adic expansion of λ,
where each λi is p-restricted and dominant. Then
L(λ) ∼= L(λ0)⊗ L(λ1)
(1) ⊗ . . .⊗ L(λr)
(r).
7
This theorem allows the study of weights of irreducible rational representations of
simple algebraic groups over an algebraically closed field of characteristic p to be reduced
to the study of p-restricted weights of such representations. It does not, however, allow
us to focus exclusively on p-restricted highest weights λ, since L(λ) may have dominant
weights lying below λ in the partial ordering which are not p-restricted.
If Φ has two root lengths, let X(T )S denote the set of weights λ ∈ X(T ) such that
〈λ, α〉 = 0 for all long roots α ∈ Φ. Thus the weights in X(T )S are Z-linear combinations
of fundamental weights corresponding to short simple roots, i.e., the set of weights with
short support. We analogously define X(T )L.
In Chapters 6 and 7 we shall make use of the following refinement of Theorem 2.3.
Theorem 2.4 (Steinberg, [25]). Let p = 2 if G is of type Bn, Cn or F4 and p = 3 if G
is of type G2. If λ ∈ X(T ) is a dominant weight then λ = µ + ν for unique dominant
weights µ ∈ X(T )L and ν ∈ X(T )S and L(λ) ∼= L(µ)⊗ L(ν).
We now state the important Lang-Steinberg theorem [1, §1.17] which shall be required
in the next chapter.
Theorem 2.5. Let G be a connected algebraic group over K = Fp. If F : G → G is
a surjective homomorphism such that GF is finite then the map L : G → G defined by
L(g) = g−1F (g) is surjective.
2.2 Premet’s theorem
We begin this section with a definition. Assume that the characteristic of K is positive.
A rational K-representation ρ of G is called infinitesimally irreducible if its differential
dρ defines an irreducible representation of the Lie algebra g.
It has been proven by Curtis in [7] that a rational representation ρ is infinitesimally
irreducible if and only if its highest weight λ is p-restricted.
Let e(Φ) denote the maximum of the squares of the ratios of the lengths of the roots
in Φ, so e(Φ) is 2 if Φ is of type Bn, Cn or F4, 3 if Φ is of type G2 and 1 otherwise.
8
Theorem 2.6 (Premet, [20]). Suppose that p > e(Φ). Then the set of weights of an
infinitesimally irreducible representation ρ of G with highest weight λ coincides with the
set of weights of an irreducible complex representation ρC of g with the same highest
weight.
Consequently, the set of weights of the irreducible module L(λ) is the union of the
Weyl group orbits of dominant weights µ with µ 6 λ provided that p > e(Φ).
The theorem proved a conjecture of I.D. Suprunenko who had shown that the result
holds for groups of type Al, l > 1 and for a few other special cases.
The cases in which p 6 e(Φ) certainly need to be excluded. There are numerous
counterexamples as we shall see later in Theorem 2.7. The theorem tells us that, provided
p > e(Φ), the weights of L(λ) are the same as the weights of the irreducible representation
with highest weight λ of a Lie algebra of the same type as G.
Let G be a simple algebraic group of simply connected type over Fp. Lu¨beck has
determined (using computational methods) all irreducible G-modules parameterised by
p-restricted highest weights in defining characteristic of dimension bounded by a given
integer M ; details are given in [19]. He details for groups of small rank (An, 2 6 n 6 20;
Bn, 2 6 n 6 11; Cn, 3 6 n 6 11; Dn, 4 6 n 6 11) not only the parameterising highest
weights and the dimensions of the modules, but also the multiplicities of the dominant
weights lower than the highest weight in the partial ordering. For these groups of small
rank the bound is given explicitly in Table 1 of [19] and for groups of large rank,M = n
3
8
if
G is of type An and M = n
3 otherwise. The information is accessible on Lu¨beck’s website
[18]. This explicit information about highest weight modules of bounded dimension in all
characteristics is crucial to our work for groups of small rank and the value of M in each
case is usually more than sufficient for our requirements.
Lu¨beck’s paper is an extension of work by Gilkey and Seitz in [9] where they considered
the exceptional types of Lie algebras and algebraic groups.
For almost all primes p, the weight multiplicities and the dimension of an irreducible
module for an algebraic group defined over a field of characteristic p > 0 are the same
as those of the corresponding simple algebraic group in characteristic 0. This will allow
9
us to state our results for characteristic 0 rather than solely for the modular case when
p > 0. We note that in characteristic 0 the well-known formula of Freudenthal [13, §22.3]
can be used to compute weight multiplicities.
Consider the action of G on L(λ) for some λ ∈ X+. Fix α ∈ Φ, and take some
µ ∈ Π(λ) where Π(λ) denotes the multiset of weights of L(λ). The weights in Π(λ) of the
form µ+ iα, i ∈ Z form a string µ− rα, . . . , µ, . . . , µ+ qα called the α-string through µ.
The Weyl group element wα reverses the string and r − q = 〈µ, α〉. We call such a string
a weight string and we refer the reader to [13, §21.3] for more details. A weight string of
length k is one consisting of k weights. We say that Π(λ) is saturated if for all µ ∈ Π(λ)
and α ∈ Φ the weight µ− iα ∈ Π(λ) for each i between 0 and 〈µ, α〉. In the cases where
the conclusion of Premet’s theorem does not hold, the weight strings that are present are
not necessarily saturated. The weights that are not present lie in the Weyl group orbit of
a dominant weight which is not present for a certain characteristic. As an example, for
type Bn the Weyl group orbit of the weight ω2 for the module L(ω3) is not present when
the characteristic of K is 2.
At times we shall employ a generalisation of the notion of a weight string. Given
two or more orthogonal roots, we take the union of their weight strings; each connected
component of the resulting graph is called a weight net. For example, if (G, λ) = (A5, ω3)
then the following is a 2 × 2 weight net; horizontally there are α1-strings and vertically
α4-strings. Note that we have omitted a factor of
1
5
on each coefficient.
2 -1 1 3 -3 -1 1 3
2 -1 1 -2 -3 -1 1 -2
More briefly, rather than state each weight in the weight net above, we use the notation
· -1 1 ·. In general, we shall use dots to stand for any integers that make an expression a
weight in Π(λ).
10
2.3 Zalesski’s theorem
A recent expository paper of Zalesski provides more precise information in the cases where
the conclusion of Premet’s theorem does not hold.
Theorem 2.7 (Zalesski, [29]). Let G be a simple algebraic group defined over a field
of characteristic p and φ a tensor indecomposable p-restricted G-module. Let µ be the
highest weight of φ. Then the weights of φ are the same as the weights of an irreducible
representation of a complex Lie algebra of the same type as G unless one of the following
holds:
(i) p = 2 and G is of type Bn;
(ii) p = 2, G is of type Cn and µ = ωn;
(iii) p = 2, G is of type G2 and µ = ω1;
(iv) p = 3, G is of type G2 and µ = ω2 or 2ω2;
(v) p = 2, G is of type F4 and µ = ω1, ω2 or ω1 + ω2.
The proof uses the tables of Gilkey and Seitz in [9] for the exceptional types. For type
Cn with p = 2, Zalesski views weights with short support in Cn as weights inside Dn. We
see that if G is of type Bn for p = 2 there is no improvement on Premet’s theorem. We
shall make use of (ii) above in Section 7.2.
The next theorem due to Seitz tells us precisely which modules are tensor indecom-
posable.
Theorem 2.8 (Seitz, [23]). Let V be a finite-dimensional vector space defined over an
algebraically closed field of characteristic p and X a simple closed connected irreducible
subgroup of SL(V ). Then V can be expressed as the tensor product V = V1 ⊗ V2 of two
non-trivial p-restricted X-modules if and only if V is a p-restricted X-module and the
following conditions hold:
11
(i) X has type Bn, Cn, F4 or G2 with p = 2, 2, 2, 3 respectively;
(ii) V1, V2 may be arranged such that Vi has highest weight λi with λ = λ1 + λ2 and λ1
(respectively λ2) has support on those fundamental dominant weights corresponding
to short (respectively long) roots.
So, for type Cn with p = 2 all 2-restricted tensor decomposable modules have the form
L(ν)⊗ L(ωn) where ν ∈ X(T )S is a 2-restricted dominant weight with short support.
2.4 Centralisers of semisimple elements
An algebraic group G acts naturally on itself by conjugation via Intx : G → G, y 7→
xyx−1. Denote the differential of this map by Adx; this is an automorphism of the
Lie algebra Lie(G) = g. The map Ad : G → Aut g ⊂ GL(g) is called the adjoint
representation of G. It can be shown that Ad : GL(n,K) → GL(n2, K) is a morphism
of algebraic groups. In fact Adx is conjugation by x when G is a closed subgroup of
GL(n,K). For a ∈ g and x ∈ G recall the following definitions:
CG(a) = {g ∈ G | Ad g.a = a},
Cg(a) = {b ∈ g | [b, a] = 0},
CG(x) = {g ∈ G | xgx
−1 = g}, and
Cg(x) = {b ∈ g | Adx.b = b}.
The global and infinitesimal centralisers of elements in the Lie algebra are related, specif-
ically the inclusion Lie(CG(a)) ⊂ Cg(a) for a ∈ g always holds and there is equality in the
case that a is a semisimple element.
As we shall see later, it will be important to consider centralisers in the algebraic
group of semisimple elements. The most suitable references for this theory are [14, §2]
and [1, §3.5].
Since we are assuming that our group G is simply connected, by a result of Steinberg
[26, §8] we see that the centraliser in G of any semisimple element is connected.
12
Proposition 2.9. Let G be a simply connected simple algebraic group, s a semisimple
element and T a maximal torus of G containing s. Then CG(s) = 〈T,Xα | α(s) = 1〉.
Thus, up to conjugation, there are only finitely many different centralisers of semisim-
ple elements in G. Moreover CG(s) is reductive with root system Φs = {α ∈ Φ | α(s) = 1}
and Weyl group Ws = 〈wα | α ∈ Φs〉.
Recall that a prime p is said to be bad for G if p divides a coefficient ai of some root
α ∈ Φ where α =
∑
αi∈Π
aiαi is expressed as a linear combination of simple roots. We say
that a prime p is good for G whenever it is not bad. It is easy to see that for types Bn,
Cn and Dn the prime p = 2 is bad, for types G2, F4, E6 and E7 the primes p = 2, 3 are
bad and for type E8 the primes p = 2, 3, 5 are bad. There are no bad primes for type An.
Deriziotis [8] found an elegant characterisation of centralisers of semisimple elements.
The possible Dynkin diagrams of the semisimple part of the centraliser of a semisimple
element are, except for a subset of the bad primes, given precisely by the possible sub-
diagrams of the extended Dynkin diagram of G (which is formed from Π˜ = Π ∪ {−α0},
where α0 is the highest root). There are bad primes for which some of these semisim-
ple centralisers do not occur; a precise method for determining these is given in work
of Hartley and Kuzucuog˘lu [11]. The possible diagrams correspond to all root systems
Ψ ⊂ Φ which have a basis which is conjugate under the Weyl group to a proper subset
of Π˜. For example, when G is of type An, taking subdiagrams of the extended Dynkin
diagram we see that centralisers of semisimple elements are of type An1 · · ·AnrTl where
n1 > · · · > nr > 0 and l +
∑
ni = n. We reserve the letter X to denote the type of the
root system of the centraliser of a semisimple element. So if n = rankT and s is regular,
i.e., dimCG(s) = n, which is the least possible dimension, we say that X = ∅.
We remark that it is necessary to distinguish between the two possible subsystems
consisting of two single non-adjacent nodes of the extended Dynkin diagram of types
Bn and Dn: the subsystem D2 formed from nodes corresponding to αn−1 and αn (or,
alternatively α1 and −α0) and the subsystem A
2
1 as before. We make the distinction
between the subsystems A3 and D3 in types Bn and Dn whenever necessary. Also for
type Bn we denote a subsystem of rank one by either A1 or B1 in order to distinguish
13
between long roots whereby Φs = {±αi} for some i 6= n or short roots {±αn}. Similarly
for type Cn a subsystem of rank one is denoted by either A1 or C1 in order to distinguish
between short roots whereby Φs = {±αi} for some i 6= n or long roots {±αn}.
2.5 Lie algebras and a result on nilpotent orbits
We shall consider the adjoint representation in the next chapter and therefore we need to
recall relevant material from the theory of Lie algebras and state part of a result about
nilpotent orbits in good characteristic due to Premet.
The following material is contained in [2] and [12]. Let Lie(G) = g be the Lie algebra
of the simple algebraic group G. Recall the Cartan decomposition
g = Cg(T )⊕
∑
α∈Φ
gα,
where Cg(T ) is a Cartan subalgebra and gα = {x ∈ g | Ad t(x) = α(t)x, t ∈ T}. Using the
same notation as in [2, Chapter 3] we let {hα, α ∈ Π; eα, α ∈ Φ} denote the Chevalley
basis for g where hα ∈ Cg(T ), [hα, eβ] = Aα,βeβ and Aα,β =
2(α,β)
(α,α)
; each element eβ is the
root vector corresponding to the root β.
Assume that g admits a non-degenerate G-invariant trace form and that p is a good
prime for the root system Φ of G. In [21, Theorem A], Premet states that for any
nilpotent element n ∈ g there exists a one-parameter subgroup λ ∈ Hom(Gm, G) such
that λ(c).n = c2n, i.e., there is a λ such that n lies in its 2-weight space g(λ, 2) = {x ∈
g | Adλ(c).x = c2x for all c ∈ K∗}.
2.6 Main theorem
We shall henceforth set
E =
⋃
s∈Gss\Z
⋃
γ∈K∗
Vγ(s),
14
where Vγ(s) = {v ∈ V | sv = γv}, Gss denotes the set of semisimple elements in G and
Z = Z(G) denotes the centre of G. We shall list all modules up to duality. This thesis
will be concerned with proving the theorem below.
Main Theorem Let G be a simple simply connected algebraic group acting on an
irreducible G-module V . If dimV 6 dimG+ 2 then
dimE = dimV,
with the following possible exceptions:
G V n
An L(3ω1) 2 (p > 3)
L(ω3) 7
Bn L(ω1 + ω2) 2 (p = 5)
L(ω6) 6
Cn L(ω2) [3,∞)
L(ωn) [5, 6] (p = 2)
L(ω3) 3 (p 6= 2)
Dn L(ω7) 7
and if dimV > dimG+ 2 then
dimE < dimV,
with the following possible exceptions:
G V n G V n
An L(ω3) 8 Cn L(ω3) 4 (p = 3)
L(ω4) 7 L(ω4) 4 (p 6= 2)
L(2ω2) 3 (p > 2) L(ω1)⊗ L(ω1)
(q) [3,∞)
Bn L(2ω1) [2,∞) (p > 2) Dn L(2ω1) [4,∞) (p > 2)
L(ω1)⊗ L(ω1)
(q) [2,∞) L(ω7) 8
L(ω8) 8
L(ω1)⊗ L(ω1)
(q) [4,∞)
The Main Theorem follows from the results in Chapter 3 if dimV 6 dimG + 2, and
Theorems 5.1, 6.1, 7.1, 8.1 and 10.1 and Lemmas 9.1, 9.2 and 9.3 if dimV > dimG + 2.
15
The proof will be split depending on the dimension of the irreducible G-module, the type
of the algebraic group involved and whether the irreducible module on which it acts is
parameterised by a p-restricted highest weight or not. In Chapter 3 we deal exclusively
with modules of dimension at most dimG + 2; in all later chapters we consider larger
modules. In Chapters 5-8 we take in turn G to be one of the classical types An, Bn, Cn
and Dn and in Chapter 9 we consider together the exceptional types E6, E7, E8, F4 and
G2. In these chapters we assume that each irreducible G-module is parameterised by a
p-restricted highest weight. In Chapter 10 we deal with the remaining modules for all
types of simple algebraic groups. These modules are parameterised by highest weights
that are not p-restricted; however by Steinberg’s tensor product theorem 2.3 they are
tensor products of modules with p-restricted highest weights together with a twist.
Since the fixed point spaces CV (s) are precisely the 1-eigenspaces V1(s) we have, by
inclusion,
dim
⋃
s∈Gss\Z
CV (s) 6 dim
⋃
s∈Gss\Z
⋃
γ∈K∗
Vγ(s)
and hence we obtain the following corollary.
Corollary 2.10. If G and V are as before and dimV > dimG + 2 then there exists a
vector v ∈ V such that StabG\Z(v) ⊂ Gu with the same possible exceptions as above, where
Gu denotes the set of unipotent elements in G.
We remark that if dimV < dimG then, by the orbit-stabiliser theorem, any vector
must have a non-trivial stabiliser in G, and so must lie in the fixed point space of some
element of G. Thus if one wished to show that there exists a regular orbit in the action
of G on V one must assume that dimV > dimG.
16
Chapter 3
Small modules
In this chapter we shall consider small modules, i.e., irreducible p-restricted G-modules of
dimension at most dimG+2 where G is a simple simply connected algebraic group defined
over an algebraically closed field. We prove general results for modules V of dimension
less than the dimension of a Borel subgroup B of G and for adjoint modules. The only
modules with dimension dimG + 1 are (G, λ) = (A1, 3ω1) with p > 3, (A3, ω1 + ω2)
with p = 3 and (An, ω1 + qω1) for n > 1. The modules with dimension dimG + 2 are
(G, λ) = (A1, 4ω1) with p > 3, (A2, 3ω1) with p > 3 and (B2, ω1 + ω2) with p = 5. We
may therefore focus on modules with dimension strictly between dimB and dimG+ 2.
3.1 General results
We shall begin by considering modules of dimension less than a Borel subgroup of G.
Lemma 3.1. Let G be a simple simply connected algebraic group, B a Borel subgroup
of G and V an irreducible G-module. If dimV < dimB then
⋃
s∈Gss\Z
CV (s) = V , and
moreover E = V .
Proof. By the orbit-stabiliser theorem, for any vector v ∈ V we have
dimStabG(v) = dimG− dimG.v > dimG− dimV > dimG− dimB = dimU
17
where U = Ru(B). Thus it is not possible for StabG(v) to be conjugate to a subgroup of
U.Z. It follows that the stabiliser in G of any vector contains a non-central semisimple
element and hence each vector lies in the fixed point space of a non-central semisimple
element. Thus we have
⋃
s∈Gss\Z
CV (s) = V , whence E = V .
It clearly follows that the conclusion of our Main Theorem holds for G-modules V
satisfying dimV < dimB since E = V .
The next proposition focuses on the adjoint action of an algebraic group on its asso-
ciated Lie algebra by conjugation. We provide a complete list of the adjoint modules for
each type in the table below.
Type Module n Type Module
An L(2ω1) 1 (p 6= 2) E6 L(ω2)
L(ω1 + ωn) [2,∞) E7 L(ω1)
Bn L(2ω2) 2 (p 6= 2) E8 L(ω8)
L(ω2) [3,∞) F4 L(ω1)
Cn L(2ω1) [3,∞) (p 6= 2) G2 L(ω2)
Dn L(ω2) [4,∞)
Proposition 3.2. Let G be a simple simply connected algebraic group acting on its irre-
ducible adjoint module V . Then E is dense in V , and moreover when p is good for G we
have E = V .
Proof. We begin without any conditions on the characteristic of the field p. Let a be a
regular semisimple element of g. The set of such elements is dense in g. Let T ′ be a
maximal torus with a ∈ Lie(T ′). Recall from Section 2.4 that global and infinitesimal
centralisers coincide for semisimple elements, so
Lie(T ′) = Cg(a) = Lie(CG(a)),
whence CG(a)
◦ = T ′. Taking s ∈ T ′ \Z we see that a is fixed by a non-central semisimple
element. Thus all regular semisimple elements of g lie in the fixed point space in g of
some s ∈ Gss \Z. Hence, by passing to the quotient of g if necessary, E is dense in V . We
18
note that the adjoint modules for F4 with p = 2 and for G2 with p = 3 are small modules
having dimension 26 and 7, respectively.
Assume from now on that p is good for G and let v ∈ g. The Lie subalgebra 〈v〉 ⊂ g
is soluble, so lies in a maximal soluble subalgebra which we may take to be b = Lie(B)
by applying an element of G. Let the Jordan decomposition of v be v = t+ n where t is
semisimple, n is nilpotent and [t, n] = 0. We may assume without loss of generality that
t ∈ t = Lie(T ) since any semisimple element of b lies in a Cartan subalgebra of b, all of
which lie in the same G-orbit. Hence we may write t =
∑
α∈Π λαhα with each λα ∈ K.
Define Ψ = {β ∈ Φ | [t, eβ] = 0} so that β ∈ Ψ if and only if
0 =
∑
α∈Π
λα[hα, eβ] =
∑
α∈Π
λαAα,β.
It is clear that Aα,β1+β2 = Aα,β1 + Aα,β2 so Ψ is a closed subsystem of Φ. Also n may be
written as a linear combination of those eβ with β ∈ Ψ since [t, n] = 0.
We take the two cases Ψ ( Φ and Ψ = Φ separately. In the former case we take a
maximal subsystem Ψ′ ⊂ Φ containing Ψ. Up to conjugacy we may assume that Ψ′ is
obtained by removing at least one node from the extended Dynkin diagram of Φ. Recall
from Section 2.4 that Ψ′ is the root system of the centraliser of a semisimple element
s ∈ G. We have that s is non-central by the assumption on Ψ and s.v = s.t + s.n = v,
i.e., v ∈ CV (s) = V1(s) so v ∈ E.
Thus we are left with the case Ψ = Φ; here t ∈ Z(g) since t commutes with t and
each root vector. Since the irreducible module is V = g/Z(g) by passing to this quotient
of g we see that v is nilpotent. As remarked in Section 2.5, we know that there is a
one-parameter subgroup λ : K∗ → G such that for all c ∈ K∗ we have λ(c).v = c2v.
Therefore, choose c ∈ K∗ such that s = λ(c) is non-central, then v ∈ Vc2(s) so v ∈ E.
We remark that if p is a bad prime for G then we need not have E = V . We may take
for example G = C2(K) with p = 2. The corresponding Lie algebra g of type C2 consists
19
of all 4× 4 matrices T satisfying T tA+ AT = 0 where
A =
(
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
)
.
Consider the matrix
v =
(
0 0 0 1
0 1 1 0
0 0 1 0
0 0 0 0
)
;
it is straightforward to check that v ∈ g. If MvM−1 = λv for some
M =
(
a b c d
e f g h
i j k l
m n p q
)
∈ G
and λ ∈ K∗ then
Mv =
(
0 b b+c a
0 f f+g e
0 j j+k i
0 n n+p m
)
= λ
( m n p q
e+i f+j g+k h+l
i j k l
0 0 0 0
)
= λvM.
Thus we must have λ = 1 and a = q 6= 0, f = k 6= 0 and d, g ∈ K; all other entries of M
are zero. Since
M ∈ C2(K) = {X ∈ GL4(K) | X
tJX = J},
where J =
(
0 K2
−K2 0
)
and K2 = ( 0 11 0 ) we have
J =
(
a 0 0 0
0 f 0 0
0 g f 0
d 0 0 a
)(
0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0
)(
a 0 0 d
0 f g 0
0 0 f 0
0 0 0 a
)
=
(
0 0 0 a2
0 0 f2 0
0 −f2 0 0
−a2 0 0
)
.
Therefore a2 = f 2 = 1, i.e., a = f = 1 and M is unipotent, hence there is no non-central
semisimple element of G for which v is an eigenvector, i.e., v /∈ E.
If we exclude the adjoint modules, we see from [18] that we need to examine the
following modules V with dimB 6 dimV 6 dimG: for type An the modules L(2ω1) for
n ∈ [2,∞) with p > 2, L(ω3) for n ∈ [5, 7] and L(ω1) for n = 1; for type Bn the module
L(ωn) for n ∈ [5, 6]; for type Cn the modules L(ω2) for n ∈ [3,∞), L(ω3) for n = 3 with
p 6= 2 and L(ωn) for n ∈ [5, 6] with p = 2; for type Dn the module L(ωn) for n = 7. For
the exceptional groups we observe from Lu¨beck’s tables [18] that there are no irreducible
20
modules V with dimB 6 dimV 6 dimG other than the adjoint modules.
3.2 Twisted modules
In the next proposition we exploit the fact that L(ω1) is the natural module for G of type
An to deal with L(ω1) ⊗ L(ω1)
(q) and L(ω1) ⊗ L(ωn)
(q). These modules have dimension
dimG+1. Fix a basis of L(ω1) and denote it by v1, . . . , vm wherem = n+1 = dimL(ω1) =
dimL(ωn).
Proposition 3.3. Let G be a simple algebraic group of type An with n > 1 acting on
V where we take V to be either L(ω1) ⊗ L(ω1)
(q) or L(ω1) ⊗ L(ωn)
(q). Then there is a
non-empty open set in V each of whose vectors has a non-central semisimple element in
its stabiliser; in particular, dimE = dimV .
Proof. We need to treat the two twisted modules in this proposition separately; the action
of SLm(K) is different in each case. For any A ∈ G the action on L(ω1) is given by
A : v 7→ Av, while the action on L(ωn) is given by A : v 7→ (A
−1)tv.
The action of A ∈ G on L(ω1)⊗L(ω1)
(q) is given as follows: v⊗w 7→ Av⊗A(q)w, i.e.,
∑
i,j
γijvi ⊗ vj 7→
∑
i,j
γij
(
Avi ⊗ A
(q)vj
)
=
∑
i,j,k,l
γijaika
q
jlvk ⊗ vl,
where Avi =
∑
k aikvk and A
(q)vj =
∑
l a
q
jlvl. Let Γ = (γij)
m
i,j=1 so the action is given
by Γ 7→ AtΓA(q). In order to show that each Γ has a non-trivial stabiliser it suffices to
show that this is true for all Γ in a non-empty open set. Thus assume that each Γ is
non-singular.
Consider for the moment the map GLm(K)→ GLm(K) given by Γ 7→ A
−1ΓA(q); this
is the action of GLm(K) on itself by F -conjugation where F is the standard Frobenius
map. Given a non-singular matrix Γ, by the Lang-Steinberg theorem 2.5, there exists
B ∈ GLm(K) such that Γ = B
−1F (B) = B−1IF (B). Thus Γ lies in the same F -
conjugacy class as the identity I. The stabiliser of I in the F -conjugation action consists of
21
the F -stable non-singular matrices GLm(q). Since the stabiliser of I contains non-central
semisimple elements, so does that of Γ. By replacing F : A 7→ A(q) by the Frobenius map
Fτ where τ : A 7→ (A−1)t we can similarly deal with the relevant map Γ 7→ AtΓA(q). If
we instead consider the action on L(ω1) ⊗ L(ωn)
(q) then the stabiliser of I under Fτ is
GUm(q) which contains non-central semisimple elements, though not necessarily lying in
G. By taking elements of the conjugate of GUm(K) which have determinant 1, we see
that the stabiliser of Γ does indeed contain non-central semisimple elements of G.
3.3 Finite orbit modules
Let G be a connected linear algebraic group defined over an algebraically closed field K
of characteristic p > 0 and let V be a finite-dimensional irreducible rational G-module.
We shall say that a module is a finite orbit module if G has only a finite number of orbits
on the set of vectors in V . These are classified in [10, Theorem 1] by Guralnick, Liebeck,
Macpherson and Seitz. A G-module V is called a prehomogeneous space if G has an open
dense orbit on V . It is clear that a finite orbit module is a prehomogeneous space. If G
is simple then, by [10, Corollary 1], the converse is also true. A finite orbit module V
satisfies dimV 6 dimG by the orbit-stabiliser theorem, hence is a small module in our
sense.
We can use existing literature to obtain the number of orbits, orbit representatives
and (useful for our work) the point stabilisers for some of these finite orbit modules.
The irreducible prehomogeneous spaces in characteristic 0 were classified by Sato and
Kimura [22], and were classified for positive characteristic by Chen [3]. Chen found that
nearly all irreducible prehomogeneous spaces in characteristic p > 0 were obtained from
the corresponding vector space in characteristic 0 by reducing modulo p. One of the
exceptions was the GL4(K)-module L(ω1+ω2) for p = 3 and this 16-dimensional module
was studied in [4], and subsequently [6]. By combining the information in both of these
papers we have representatives for the ten orbits in the action, as well as the structure of
the point stabilisers of these representatives. By inspection, each vector in L(ω1+ω2) for
22
p = 3 is stabilised by a non-central semisimple element in SL4(K).
The spin module L(ω5) for B5 is a finite orbit module. If p 6= 2 then Igusa details
in [15, Proposition 6] the five orbits that occur and the point stabilisers. If p = 2 then
in fact there are six orbits as detailed in [10, Lemma 2.11]. The information provided in
these references allows us to conclude that each vector in the Spin11(K)-module L(ω5) is
stabilised by a non-central semisimple element.
Thus we have:
Lemma 3.4. Let G be a simple simply connected algebraic group and V = L(λ) an
irreducible G-module. If (G, λ) = (A3, ω1 + ω2) with p = 3 or (B5, ω5) then we have
E = V .
We note that the G-module V = L(ω3) for G = An with n ∈ [5, 7] is a finite orbit
module. We shall show that E = V for the cases n = 5 in 3.4 and n = 6 in 3.6; in the
latter case point stabilisers have been determined by Chen, Cohen and Wales so we can
conclude as above. Some of the possible exceptions for small modules listed in the Main
Theorem such as L(ω3) for A7, L(ω3) for C3 with p 6= 2 and L(ω7) for D7 are finite orbit
modules. We have not been able to find information in the literature to assist with these
cases.
3.4 Two specific cases
In this section we shall consider two irreducible G-modules where G is of type An. We
show that both of these modules are the union of eigenspaces of non-central semisimple
elements.
Lemma 3.5. Let G be a simple simply connected algebraic group of type An acting on
the irreducible module V = L(2ω1) for n > 2 with p 6= 2. Then E = V .
Proof. The irreducible module with highest weight 2ω1 is S
2N , the symmetric square of
23
the natural module N = L(ω1). For any 0 6= x ∈ S
2N we may write
x =
m∑
i,j=1
cijvi ⊗ vj
where m = n + 1, cij = cji for i 6= j and v1, . . . , vm is the standard basis of N . Let
S = StabG(x) and assume that S contains no non-central semisimple element. We may
assume that S ⊂ U.Z by conjugating S. Since dimS2N = 1
2
(n + 1)(n + 2) we have
dimS > 1
2
(n − 1)(n + 2) = dimU − 1 by the orbit-stabiliser theorem. Thus for any
subgroup H of U we have dim(S ∩H) > dimH − 1. In particular, if we take H = XγXδ
where γ = α2 + · · · + αn and δ = α1 + γ then dimH = 2; we have dim(S ∩ H) > 1 so
that S contains a non-trivial element g = xγ(a)xδ(b) with (a, b) 6= (0, 0). It is clear that
gvi = vi for i < m and gvm = vm + bv1 + av2. Since g ∈ S and g fixes each vi ⊗ vj with
i, j < m we see that g must fix the vector
y =
m−1∑
i=1
cim (vi ⊗ vm + vm ⊗ vi) + cmmvm ⊗ vm.
By inspecting the coefficients in
gy − y =
m−1∑
i=1
cim [(av2 + bv1)⊗ vi + vi ⊗ (av2 + bv1)]
+ cmm[a(v2 ⊗ vm + vm ⊗ v2) + b(v1 ⊗ vm + vm ⊗ v1)
+ a2(v2 ⊗ v2) + ab(v1 ⊗ v2 + v2 ⊗ v1) + b
2(v1 ⊗ v1)]
of vk ⊗ vi + vi ⊗ vk for 2 < i < m and k ∈ {1, 2} we see that cim = 0. Analogously by
inspecting the coefficients of vk ⊗ vm + vm ⊗ vk for k ∈ {1, 2} we find cmm = 0. Now
0 ≡ gy − y = 2bc1m(v1 ⊗ v1) + 2ac2m(v2 ⊗ v2) + (ac1m + bc2m)(v1 ⊗ v2 + v2 ⊗ v1)
so that 2bc1m = 2ac2m = ac1m+ bc2m = 0 and since (a, b) 6= (0, 0) we have c1m = c2m = 0.
Thus cim = cmi = 0 for 1 6 i 6 m, whence x =
∑m−1
i,j=1 cijvi⊗ vj. If s = diag(ν, . . . , ν, ν
−n)
24
for some ν ∈ K∗ with νm 6= 1 then s is a non-central semisimple element with sx = ν2x,
i.e., x is an eigenvector with eigenvalue ν2.
Lemma 3.6. If G = SL6(K) acts on the irreducible module V = L(ω3) then E = V .
Proof. The irreducible module L(ω3) is
∧3N , the third exterior power of the natural
module. If n = 5 then dimL(ω3) = dimB = 20 = codimU . Take any
0 6= x =
∑
i<j<k
λijkvi ∧ vj ∧ vk ∈
∧3
N
with λijk ∈ K and 1 6 i < j < k 6 6, and set S = StabG(x). Assume that S contains no
non-central semisimple element. We have dimU 6 dimS by the orbit-stabiliser theorem
so we may assume that U 6 S by taking conjugates of S, hence gx = x for any g ∈ U .
Let β1 = α1+· · ·+α5, β2 = α2+· · ·+α5 and β3 = α3+α4+α5, and consider the action
of xβi(1) for each i on the standard basis; it sends vj 7→ vj for j 6= 6 and v6 7→ v6 + vi.
Thus we find
0 ≡ xβ1(1)x− x =
∑
26i<j65
(λij6vi ∧ vj ∧ v1)
in which case all coefficients λij6 with 2 6 i < j 6 5 are zero. Similarly, by applying
xβ2(1) we see that λ1i6 = 0 for 2 < i 6 5 and finally applying xβ3(1) we see that
λ126 = 0. Now x is an eigenvector with eigenvalue ν
3 for the non-central semisimple
element s = diag(ν, ν, ν, ν, ν, ν−5) for ν6 6= 1.
3.5 Type A1
We need to consider the action of SL2(K) on the modules L(ω1), L(3ω1) and L(4ω1).
Lemma 3.7. If G = SL2(K) acts on the irreducible module V = L(ω1) then E = V .
Proof. The module L(ω1) is the natural module N . If n = 1 then dimG = 3 and dimB =
2. Consider 0 6= x = av1 + bv2 ∈ N and let S = StabG(x). Suppose that S contains
no non-central semisimple element then, as before, we can assume that U = Xα1 6 S
25
by conjugating S and so moving x within its orbit. Since xα1(1) sends v1 7→ v1 and
v2 7→ v2 + cv1 and xα1(c)x − x = 0 we must have b = 0. Then x is an eigenvector with
eigenvalue ν for s = diag(ν, ν−1) ∈ Gss \ Z for ν 6= ±1.
The module L(rω1) for A1 with r > 1 can be viewed as the space of homogeneous
polynomials of degree r in two variables x and y. The action of SL2(K) is given by the
following: 
a b
c d

 .(xiyj) = (ax+ cy)i(bx+ dy)j
extended by linearity.
Lemma 3.8. If G = SL2(K) acts on the irreducible module V = L(rω1) for r ∈ 3, 4 then
dimE = dimV .
Proof. If r = 4, there are five basis vectors x4, x3y, x2y2, xy3 and y4. The semisimple
element s = ( i 00 −i ) where i
2 = −1 has eigenspace V1(s) = 〈x
4, x2y2, y4〉 with eigenvalue 1.
It is clear that the centraliser CG(s) of s is the maximal torus T consisting of diagonal
matrices. Thus applying diagonal matrices preserves the eigenspace as they just scale the
basis elements. We can use the Bruhat decomposition [2, §8.4] to obtain the set of coset
representatives for T . Given such a coset representative g, the eigenspace for gsg−1 is
gV1(s). Each g is either of the form uh or uhw˙α1u
′ where
u = ( 1 t0 1 ) ∈ U, u
′ = ( 1 t
′
0 1 ) ∈ U, h =
(
µ 0
0 µ−1
)
∈ T and w˙α1 = (
0 1
−1 0 ) ∈ NG(T ),
and t, t′ ∈ K and µ ∈ K∗. In the second case, by applying uhw˙α1u
′ to ax4 + bx2y2 + cy4
we obtain an expression with five parameters a, b, c, t, t′. Thus the closure of the union of
these eigenspaces
⋃
g∼s V1(g) is five-dimensional, hence dimE = dimV .
Analogously, for r = 3 we can reach the same conclusion; here the basis vectors are x3,
x2y, xy2 and y3. Now s has two eigenspaces V−i(s) = 〈x
3, y3〉 and Vi(s) = 〈x
2y, xy2〉.
26
3.6 The action of SL7(K) on L(ω3)
In this section we consider the action of G = SL7(K) on the irreducible module V =
L(ω3). As previously mentioned, this is a finite orbit module. The nine orbits and point
stabilisers for each orbit have been precisely determined in [5, Table 1]. It is clear from
this information that E = V for this module.
We also detail an alternative approach to show that E = V due to Dr. R. Lawther.
This method is based on techniques used in the previous two sections.
Recall in general that Φ = Φ(An) = {ǫi − ǫj | 1 6 i, j 6 n + 1}. Here we shall write
roots in the form i− j rather than ǫi− ǫj, where 1 6 i, j 6 7; assume Φ
+ = {i− j | i < j}.
Let v1, . . . , v7 be the standard basis of the natural module of G. Given i, j, k 6 7 with
i 6= j 6= k 6= i, write vijk for the standard basis vector vi ∧ vj ∧ vk of V , so that vijk =
vjki = vkij = −vikj = −vkji = −vjik; we shall write the unordered triple {i, j, k} as simply
ijk. This should not cause confusion, as subscripts will never be unordered triples.
Take v ∈ V ; write v =
∑
i<j<k λijkvijk and set ∆ = ∆(v) = {ijk | λijk 6= 0} (note that
this is well-defined).
We shall show that v ∈ E; we begin with a series of lemmas showing that this conclu-
sion holds under various hypotheses on ∆. Our first two lemmas are elementary.
Lemma 3.9. If |∆| 6 6 then v ∈ E.
Proof. Take λ ∈ K∗ and p1, . . . , p7 ∈ Z, and let s = diag(λp1 , . . . , λp7); then svijk =
λpi+pj+pkvijk. Thus given q ∈ Z we have sv = λqv if and only if pi + pj + pk = q for all
ijk ∈ ∆; this imposes |∆| homogeneous linear conditions on the 8 variables p1, . . . , p7, q,
in addition to the condition p1 + · · · + p7 = 0 which guarantees s ∈ G. Thus provided
|∆| 6 6 there will be a non-zero solution; since Z(G) is finite we may choose λ so that
s /∈ Z(G).
For l 6 7 write ∆l = {ijk | i, j, k 6= l}; given l,m 6 7 write ∆l,m = ∆l ∩∆m.
Lemma 3.10. If ∆ ⊆ ∆l for some l 6 7 then v ∈ E.
27
Proof. Take λ ∈ K∗ with λ7 6= 1 and let s = diag(λp1 , . . . , λp7) where pl = −6 and pi = 1
if i 6= l; then sv = λ3v and s /∈ Z(G).
In the arguments to follow we shall frequently apply root elements or Weyl group
elements to modify v and hence ∆; for H a subgroup of G rather than repeating the
phrase applying a suitable element of H we shall abbreviate this to applying H.
Lemma 3.11. If |∆ \∆6|, |∆ \∆7| 6 1 then v ∈ E.
Proof. By Lemma 3.10 it suffices to treat the case where |∆ \∆6| = |∆ \∆7| = 1. If ∆ \
∆6 = ∆\∆7, there exists i 6 5 such that ∆ ⊆ ∆6,7∪{i67}, and then diag(1, 1, 1, 1, 1, λ, λ
−1)v =
v; so we may assume there exist i, j, k, l 6 5 such that ∆ ⊆ ∆6,7 ∪ {ij6, kl7}. If
{i, j} = {k, l}, applying X6−7 we may ensure ij6 /∈ ∆, so that ∆ ⊆ ∆6 and the re-
sult follows by Lemma 3.10; so we may assume that {i, j} 6= {k, l}. Given a 6 5 with
a 6= i, j, applying Xa−6 we may ensure ija /∈ ∆; likewise, given b 6 5 with b 6= k, l,
applying Xb−7 we may ensure that klb /∈ ∆. Now if {i, j} ∩ {k, l} = ∅, take m such
that {i, j, k, l,m} = {1, 2, 3, 4, 5}, then ∆ ⊆ {ikm, ilm, jkm, jlm, ij6, kl7} and the result
follows by Lemma 3.9; so we may assume i = k, j 6= l. Take m,n with m < n such
that {i, j, l,m, n} = {1, 2, 3, 4, 5}, then ∆ ⊆ {jlm, jln, imn, jmn, lmn, ij6, kl7}; applying
Xm−n we may ensure |∆ ∩ {jlm, jln}| 6 1 and the result follows by Lemma 3.9.
Lemma 3.12. If there exist distinct l,m, n 6 5 such that ∆ ⊆ ∆6,7∪{in6, in7 | i 6 5, i 6=
n} ∪ {lm6, lm7}, then v ∈ E.
Proof. Applying 〈w1−2, w2−3, w3−4, w4−5〉 we may assume (l,m, n) = (1, 2, 5). Applying
X6−7 and 〈w6−7〉 we may ensure 127 /∈ ∆; likewise applying X3−4 and 〈w3−4〉 we may
ensure 357 /∈ ∆, and applying X1−2 and 〈w1−2〉 we may ensure 157 /∈ ∆. Suppose
457 ∈ ∆; applying X2−4X6−7 we may ensure 257, 456 /∈ ∆, and applying X1−2X2−3X1−3
we may ensure |∆ ∩ {156, 256, 356}| 6 1. Now, by Lemma 3.11, we may assume 126 ∈ ∆
and there exists i 6 3 with ∆ ∩ {156, 256, 356} = {i56}. If i = 3, applying
X5−6X3−6X2−6X1−6X3−7X2−7X1−7
28
we may ensure 125, 123, 235, 135, 345, 245, 145 /∈ ∆, so ∆ ⊆ {124, 134, 234, 126, 356, 457}
and the result follows by Lemma 3.9; if instead i ∈ {1, 2}, applying
X5−6X4−6X3−6X3−7X2−7X1−7
we may ensure 125, 124, 123, 345, 245, 145 /∈ ∆, so ∆ ⊆ {134, 234, 135, 235, 126, i56, 457}
and diag(λ3, λ3, λ−4, λ3, λ3, λ−4, λ−4)v = λ2v. So we may assume 457 /∈ ∆; by Lemma
3.10 we may assume 257 ∈ ∆, else ∆ ⊆ ∆7; now applying X6−7 we may ensure 256 /∈ ∆.
If 356, 456 /∈ ∆ then applying X2−5 we may ensure |∆ ∩ {126, 156}| 6 1 and the result
follows by Lemma 3.11; so we may assume ∆ ∩ {356, 456} 6= ∅, and applying 〈w3−4〉
we may assume 456 ∈ ∆. Now applying X3−4X1−4 we may ensure 356, 156 /∈ ∆. If
126 /∈ ∆ the result follows by Lemma 3.11, so we may assume 126 ∈ ∆; applying
X5−6X4−6X3−6X2−6X1−6 we may ensure 125, 124, 345, 245, 145 /∈ ∆, so ∆ ⊆ {123, 134, 234, 135, 235, 126
and diag(λ3, λ3, λ−4, λ3, λ3, λ−4, λ−4)v = λ2v
Lemma 3.13. If there exist distinct m,n 6 5 such that ∆ ⊆ ∆6,7 ∪ {im6, in6 | i 6 5, i 6=
m,n} ∪ {mn6} ∪ {in7 | i 6 5, i 6= n}, then v ∈ E.
Proof. Applying
∏
i<j65
i,j 6=m,n
Xi−j we may ensure that there exists l 6 5 with l 6= m,n such
that ∆ ∩ {im6 | i 6 5, i 6= m,n} ⊆ {lm6}, and the result follows by Lemma 3.12.
Lemma 3.14. If there exists l 6 5 such that ∆ ∩ {il6, il7 | i 6 5, i 6= l} = ∆ ∩ {i67 | i 6
5} = ∅, then v ∈ E.
Proof. Applying 〈w1−2, w2−3, w3−4, w4−5〉 we may assume l = 5; by Lemma 3.10 we may
assume ∆ * ∆7, and applying 〈w1−2, w2−3, w3−4〉 we may assume 347 ∈ ∆. Now applying
X2−3X1−3X2−4X1−4X2−7X1−7 we may ensure 247, 147, 237, 137, 234, 134 /∈ ∆; if 126, 127 /∈
∆ the result follows by Lemma 3.13 with (m,n) = (3, 4), so we may assume there exists
k ∈ {6, 7} with 12k ∈ ∆. Now applying X4−kX3−k we may ensure 123, 124 /∈ ∆, so
∆ ⊆ {ij5, ij6 | i, j 6 4} ∪ {127, 347} and diag(λ3, λ3, λ3, λ3, λ−4, λ−4, λ−4)v = λ2v.
We may now proceed to our general argument.
29
Theorem 3.15. Let G = SL7(K) act on the irreducible module V = L(ω3). Then E = V .
Proof. Recall that U =
∏
α∈Φ+ Xα and let U
′ =
∏
i<6Xi−6
∏
j<7Xj−7, so that dimU = 21
and dimU ′ = 11. Suppose StabG(v)
◦ is unipotent; applying G we may assume that
StabG(v)
◦ ⊆ U . Since dimG = 48 while dimV = 35, we must have dimStabG(v)
◦ > 13;
thus dim (StabG(v)
◦ ∩ U ′) > 3. From now on we shall assume merely that v ∈ V is such
that dim (StabG(v)
◦ ∩ U ′) > 3, and we shall show that this condition implies that v ∈ E;
this will allow us to apply 〈w1−2, w2−3, w3−4, w4−5〉 wherever it is convenient to do so, since
this subgroup of W preserves U ′.
Order the roots of U ′ as
5− 6 ≺ 4− 6 ≺ 3− 6 ≺ 2− 6 ≺ 1− 6 ≺ 6− 7 ≺ 5− 7 ≺ 4− 7 ≺ 3− 7 ≺ 2− 7 ≺ 1− 7.
Given u ∈ U ′ we may write u =
∏
xα(tα). If α is the first root in the order given
for which tα 6= 0, we say that u begins with α. Let Γ = {γ ∈ Φ
+ | there exists u ∈
StabU ′(v) beginning with γ}.
Observe that if u ∈ U ′ and µ 6= 0, then u and I + µ(u − I) fix the same vectors in
V . Thus for all γ ∈ Γ we may choose gγ ∈ StabU ′(v) such that gγ = xγ(1)
∏
β≻γ xβ(tβ)
for some tβ. Moreover, if γ, γ
′ ∈ Γ and γ′ ≻ γ we may replace gγ by gγ − tγ′(gγ′ − I) to
ensure that the projection of gγ on Xγ′ is trivial; we call this adjusting gγ by gγ′ . Taking
successively the γ′ ∈ Γ with γ′ ≻ γ in increasing order, we may obtain a unique element
gγ ∈ xγ(1)
∏
β≻γ
β /∈Γ
Xβ lying in StabU ′(v). It is then clear that
StabU ′(v) =
{
I +
∑
γ∈Γ
tγ(gγ − I) | tγ ∈ K for all γ ∈ Γ
}
;
so dimStabU ′(v) = |Γ|, whence |Γ| > 3.
Thus we may take distinct γ1, γ2, γ3 ∈ Γ; given i 6 3 we shall write simply gi for gγi .
We shall consider three cases, which between them cover all possibilities:
(I) (γ1, γ2, γ3) = (a− 7, b− 7, c− 7) for distinct a, b, c;
30
(II) (γ1, γ2) = (a− 7, b− 6) for distinct a, b, and γ3 is either c− 7 for c 6= a or c− 6 for
c 6= b;
(III) (γ1, γ2, γ3) = (a− 6, b− 6, c− 6) for distinct a, b, c.
In the following we shall write [g, ijk] to mean the condition that the ijk-component
of gv − v is zero.
Suppose that (I) holds. We may assume a < b < c; applying
∏
i<a
Xi−a
∏
j<b
j 6=a
Xj−b
∏
k<c
k 6=a,b
Xk−c
we may assume g1 = xa−7(1), g2 = xb−7(1) and g3 = xc−7(1). Given distinct i, j 6 6, if
i, j 6= a then [g1, ija] which implies that ij7 /∈ ∆, if i, j 6= b then [g2, ijb] which implies
that ij7 /∈ ∆, and if i, j 6= c then [g3, ijc] which implies that ij7 /∈ ∆. Thus ∆ ⊆ ∆7 and
the result follows by Lemma 3.10.
Suppose (II) holds. Applying
∏
i<aXi−a
∏
j<bXj−b we may assume g1 = xa−7(1) and
g2 ∈ xb−6(1)
∏
j 6=aXj−7. Given i 6 5 with i 6= b we have [g2, ib7] which implies that
i67 /∈ ∆.
First suppose c < 6; then according as γ3 = c − 6 or c − 7, either [g3, bc7] or [g3, bc6]
which implies that b67 /∈ ∆, so ∆∩{i67 | i 6 5} = ∅. Given distinct i, j 6 5 with i, j 6= a,
[g1, ija] which implies that ij7 /∈ ∆. Thus, if a = 6 we have ∆ ⊆ ∆7 so the result follows
by Lemma 3.10; we may therefore assume a 6 5, and ∆ ⊆ ∆7 ∪ {ia7 | i 6 5, i 6= a}. Now
given distinct i, j 6 5 with i, j 6= a, b, we have [g2, ijb] which implies that ij6 /∈ ∆; so
∆ ⊆ ∆6 ∪ {ia6, ib6 | i 6 5, i 6= a, b} ∪ {ab6}, and the result follows by Lemma 3.13 with
(m,n) = (b, a).
So we may suppose instead c = 6, so that γ3 = 6 − 7; applying
∏
k<6Xk−6 we may
assume g3 = x6−7(1). Given distinct i, j 6 5, we have [g3, ij6] which implies that ij7 /∈ ∆;
now given distinct i, j 6 5 with i, j 6= b, we have [g2, ijb] which implies that ij6 /∈ ∆;
so ∆ ⊆ ∆6,7 ∪ {ib6 | i 6 5, i 6= b} ∪ {b67}. Thus if b67 /∈ ∆ we have ∆ ⊆ ∆7 and the
result follows by Lemma 3.10; if instead b67 ∈ ∆, applying
∏
i65
i6=b
Xi−7 we may ensure
31
∆ ⊆ ∆6,7 ∪ {b67} and the result follows by Lemma 3.11.
Finally suppose (III) holds. Applying 〈w1−2, w2−3, w3−4, w4−5〉 we may assume (a, b, c) =
(1, 2, 3). We have [g2, 127] which implies that 167 /∈ ∆, while if 2 6 i 6 5 we have
[g1, i17] which implies that i67 /∈ ∆. Thus ∆ ∩ {i67 | i 6 5} = ∅. For p 6 3 write
gp = xp−6(1)
∏
q<7 xq−7(tpq). If there exists p 6 3 with tp6 6= 0, then given distinct i, j 6 5
we have [gp, ij6] which implies that ij7 /∈ ∆, so ∆ ⊆ ∆7 and the result follows by Lemma
3.10. Therefore, we may assume that t16 = t26 = t36 = 0.
First suppose there exist p 6 3 and q ∈ {4, 5} with tpq 6= 0. Applying 〈w1−2, w2−3〉
(and permuting g1, g2, g3 accordingly) and 〈w4−5〉 we may assume t15 6= 0; applying
X4−5X3−5X2−5X1−5 and T we may assume t15 = 1 and t11 = t12 = t13 = t14 = 0, so that
g1 = x1−6(1)x5−7(1); adjusting g2 and g3 by g1 and applying X1−2X1−3 we may assume
t25 = t35 = 0. Now [g1, 123], [g1, 124], [g1, 134], [g1, 235], [g1, 245] and [g1, 345] imply that
236, 246, 346, 237, 247, 347 /∈ ∆.
Now if there exists p′ ∈ {2, 3} with tp′4 6= 0, applying 〈w2−3〉 (and permuting g2, g3
accordingly) we may assume t24 6= 0; applying X3−4X2−4X1−4 and T we may assume
t24 = 1 and t21 = t22 = t23 = 0, so that g2 = x2−6(1)x4−7(1). Now [g2, 123], [g2, 235],
[g2, 134] and [g2, 345] imply that 136, 356, 137, 357 /∈ ∆, and the result follows by Lemma
3.14 with l = 3. So we may assume t24 = t34 = 0. Now applying X1−5X6−7 so as to
preserve g1 we may assume t33 = 0. We then have [g3, 134] and [g3, 345] implying that
146, 456 /∈ ∆, whence [g1, 145] which implies that 147 /∈ ∆. Now if 457 /∈ ∆ the result
follows by Lemma 3.14 with l = 4, so we may assume 457 ∈ ∆; then [g2, 145] which
implies that t21 = 0, [g2, 245] which implies that t22 = 0, [g2, 345] which implies that
t23 = 0, so g2 = x2−6(1); [g3, 145] which implies that t31 = 0 and [g3, 245] which implies
that t32 = 0, so g3 = x3−6(1). Now [g2, 123], [g2, 125], [g2, 235], [g3, 123] and [g3, 235] imply
that 136, 156, 356, 126, 256 /∈ ∆, so ∆ ⊆ ∆6 and the result follows by Lemma 3.10.
Thus we may assume that t14 = t15 = t24 = t25 = t34 = t35 = 0.
Next suppose there exist distinct p, q 6 3 with tpq 6= 0. Applying 〈w1−2, w2−3〉
(and permuting g1, g2, g3 accordingly) we may assume t13 6= 0; applying X2−3X1−3
and T and adjusting g3 by g2 and g1 we may assume t13 = 1 and t11 = t12 = 0,
32
so that g1 = x1−6(1)x3−7(1); adjusting g2 by g1 and applying X1−2 we may assume
t23 = 0; applying X1−3X6−7 so as to preserve g1, and adjusting g3 by g1, we may as-
sume t22 = 0. Now [g1, 124], [g1, 125], [g1, 145], [g1, 234], [g1, 235] and [g1, 345] imply that
246, 256, 456, 247, 257, 457 /∈ ∆; now [g2, 124], [g2, 234], [g2, 125] and [g2, 235] imply that
146, 346, 156, 356 /∈ ∆; now [g1, 134] and [g1, 135] imply that 147, 157 /∈ ∆. Thus we
have ∆ ⊆ ∆6,7 ∪ {126, 136, 236, 127, 137, 237, 347, 357}. If 357 /∈ ∆ the result follows by
Lemma 3.14 with l = 5, so we may assume 357 ∈ ∆; but now applying X4−5 we may
ensure 347 /∈ ∆ and the result follows by Lemma 3.14 with l = 4.
Thus we may assume that t12 = t13 = t21 = t23 = t31 = t32 = 0. Applying X6−7
we may assume t33 = 0, so g3 = x3−6(1). Given distinct i, j 6 5 with i, j 6= 3, we have
[g3, ij3] which implies that ij6 /∈ ∆. If there exists p 6 2 with tpp 6= 0, applying 〈w1−2〉
(and permuting g1, g2 accordingly) we may assume t11 6= 0; now [g1, 124], [g1, 125] and
[g1, 145] imply that 247, 257, 457 /∈ ∆; so
∆ ⊆ ∆6,7 ∪ {i36 | i 6 5, i 6= 3} ∪ {i17, i37 | i 6 5, i 6= 1, 3} ∪ {137}.
Applying 〈w6−7〉 the result follows by Lemma 3.13 with (m,n) = (1, 3). Thus we may
assume t11 = t22 = 0, so g1 = x1−6(1) and g2 = x2−6(1). Hence [g2, 123], [g1, 123], [g1, 134]
and [g1, 135] imply that 136, 236, 346, 356 /∈ ∆, so ∆ ⊆ ∆6 and the result follows by
Lemma 3.10.
For type A7 the module L(ω3) remains as the only possible exception to the conclusion
of the Main Theorem for modules of type An with dimension less than dimG.
33
Chapter 4
Techniques for large modules
The aim of this chapter is to lay out the main theory which we shall apply in the later
chapters for large modules, i.e., modules V for which dimV > dimG + 2; recall that our
aim is to show that dimE < dimV . Indeed, for the remainder of this thesis we shall be
solely concerned with large modules. We shall prove that if a certain inequality dependent
on the centraliser type of a non-central semisimple element is satisfied for all such types
then the required conclusion holds. We explain the adjacency principle which is used to
obtain lower bounds on the codimension of an eigenspace. We also give criteria which we
use to obtain a relatively short list of modules which require further consideration.
4.1 Analysis of centraliser types
We begin by noting that [12, §26.2] shows that the elements of the finite centre Z =
Z(G) of G are semisimple. We shall require a division of the non-central semisimple
elements according to the type of the centraliser. Let I denote the set of possible types of
centralisers of non-central semisimple elements (see Section 2.4). By Proposition 2.9 and
Deriziotis’ theorem, there are a finite number of types X of centralisers of non-central
semisimple elements formed by taking all possible subdiagrams of the extended Dynkin
34
diagram of G. Thus I is a finite set. For each X ∈ I we set
SX = {s ∈ Gss | CG(s) of type X}
so that Gss \ Z =
⋃
X∈I SX . Hence if we can prove for each X ∈ I that
dim
⋃
s∈SX
⋃
γ∈K∗
Vγ(s) < dimV
then the conclusion of the Main Theorem holds for V .
The next result will be used repeatedly in subsequent chapters. We use the notation
dimX to denote the dimension of the centraliser of a semisimple element with root system
of type X and set codimX = dimG − dimX; also set codimVγ(t) = dimV − dimVγ(t)
for any t ∈ T . Also we write a ∼ b if and only if a, b ∈ G are conjugate in G.
We consider the following two conditions. For a fixed centraliser type X ∈ I if the
inequality
codimVγ(t) > codimX
holds for all t ∈ T ∩ SX and γ ∈ K
∗ then we shall say that
(♦) is satisfied for X.
If the inequality
codimVγ(s) > dimG
holds for all s ∈ Gss \ Z and γ ∈ K
∗ then we shall say that
(†) holds.
Certainly if the stronger condition (†) holds then condition (♦) is satisfied for all X ∈ I.
This avoids a case-by-case analysis of centraliser types.
Theorem 4.1. Let G be a simple simply connected algebraic group and let V be an
35
irreducible G-module, both defined over an algebraically closed field K. If (♦) holds for
some X ∈ I, then
dim
⋃
s∈SX
⋃
γ∈K∗
Vγ(s) < dimV.
Proof. Consider the set of weights {λi | 1 6 i 6 l} of the module (ignoring multiplicities)
ordered in some way. The set of all l× l matrices with rows and columns indexed by this
ordered set of weights whose entries are all zeros or ones is clearly finite. Given an element
s ∈ T , we define such a matrix M(s) as follows: for each pair (i, j) with i, j ∈ [1, l], we set
the (i, j)th entry of M(s) to be one if λi(s) = λj(s) and zero otherwise. In other words,
an element s ∈ T has a collection of weight spaces and the possible equality of weights
for this element is determined by the matrix M(s) of which there are only finitely many.
Now take X ∈ I and assume that (♦) holds for X. By the above, given t ∈ T ∩ SX
there are finitely many possibilities for the collection of eigenspaces of t, i.e., there exist
t1, . . . , tm ∈ T ∩ SX such that if t ∈ T ∩ SX then t has the same eigenspaces as some ti.
In particular we have the equality
⋃
s∈SX
⋃
γ∈K∗
Vγ(s) =
m⋃
i=1
⋃
s∼ti
⋃
γ∈K∗
Vγ(s).
Thus it suffices to show that for all i 6 m we have
dim
⋃
s∼ti
⋃
γ∈K∗
Vγ(s) < dimV.
Now any s ∈ ti
G has the same eigenvalues as ti, and there are only finitely many of these.
Thus by reversing the order of the two unions, we see that it suffices to show that for all
i 6 m and all γ ∈ K∗ we have
dim
⋃
s∼ti
Vγ(s) < dimV.
We claim that in fact for all t ∈ T and γ ∈ K∗ we have
36
dim
⋃
s∼t
Vγ(s) 6 dim t
G + dimVγ(t). (⋆)
Assuming that we can prove (⋆), then taking t = ti we have
dim ti
G + dimVγ(ti) = codimX + dimV − codimVγ(ti) < dimV
as required.
In order to prove (⋆) we begin by closely following the proof of [16, Proposition 1.14].
Given γ ∈ K∗, set
Yγ = {(g, v) ∈ G× V | gv = γv}.
If πγ, φ : G× V → V are the morphisms defined by
πγ(g, v) = γv, φ(g, v) = gv
then Yγ = {(g, v) ∈ G× V | πγ(g, v) = φ(g, v)}, so Yγ is a closed subvariety of G× V .
Now for t ∈ T define
Yγ,t = {(t
g, v) | g ∈ G, v ∈ V, tgv = γv}.
Then Yγ,t is a variety, and the map Yγ,t → t
G given by (tg, v) 7→ tg has fibres of dimension
Vγ(t), so
dimYγ,t = dim t
G + dimVγ(t).
In order to establish (⋆) it therefore suffices to prove
dim
⋃
s∼t
Vγ(s) 6 dimYγ,t.
Now we have the morphism π : G× V → V defined by π(g, v) = v. If we restrict π to
Yγ,t, we have a morphism whose image is
⋃
s∼t Vγ(s). Let the irreducible components of Yγ,t
37
be A1, . . . , Ak. Then for each i, π(Ai) is a closed set containing π(Ai); so
⋃k
i=1 π(Ai) is a
closed set containing
⋃k
i=1 π(Ai) = π
(⋃k
i=1Ai
)
= π(Yγ,t), and hence π(Yγ,t) ⊆
⋃k
i=1 π(Ai).
On the other hand, for all i we have π(Ai) ⊆ π(Yγ,t), so π(Ai) ⊆ π(Yγ,t) and hence⋃k
i=1 π(Ai) ⊆ π(Yγ,t). Thus π(Yγ,t) =
⋃k
i=1 π(Ai).
Now, by [24, Lemma 1.9.1(iii)], since Ai is irreducible so is π(Ai), and dimAi >
dim π(Ai). Thus
dimYγ,t = max
16i6k
dimAi > max
16i6k
dimπ(Ai) = dim
k⋃
i=1
π(Ai) = dimπ(Yγ,t) = dim
⋃
s∼t
Vγ(s)
as required to prove (⋆).
We shall prove that for a given G and irreducible module V either (†) holds or (♦)
is satisfied for all X ∈ I with a list of possible exceptions. Given X ∈ I denote by dXλ
the minimal codimension of Vγ(s) for any s ∈ SX and γ ∈ K
∗. We denote codimX =
|Φ(G)| − |Φ(X)| by eXλ . Clearly our aim when we cannot show (†) holds will be to show
that dXλ exceeds e
X
λ for all X ∈ I.
4.2 Adjacency principle
Recall that our aim for large modules is to show that dimE < dimV . We shall explain
a method which we use to enable us to determine a lower bound for the codimension of
an eigenspace Vγ(s) for any s ∈ Gss \ Z and γ ∈ K
∗.
Consider any s ∈ Gss \ Z which we may assume lies in T (by conjugation). Recall in
Proposition 2.9 that CG(s) is generated by root subgroups Xβ such that β(s) = 1 along
with the torus T . For any γ ∈ K∗ the eigenspace Vγ(s) is a sum of weight spaces. Since s
is not central there must be a root α /∈ Φs in which case µ(s) 6= (µ+α)(s) for any weight
µ and so Vγ(s) cannot contain both Vµ and Vµ+α; we shall refer to this as the adjacency
principle in later chapters. It is clear that we can naturally extend the adjacency principle
to the consideration of weight nets.
38
The following situation arises fairly often. Suppose that 〈µ, α〉 only takes the values
0 or ±1, so all weight strings are of length 1 or 2. Assume that there are k mutually
orthogonal roots α1, . . . , αk not in Φs. We see that all weight nets that occur are of the
form m1 × · · · ×mk where mi ∈ {1, 2} for each i ∈ [1, k]. Therefore, for all weight nets
other than those which are 1× · · · × 1, there exists α ∈ {α1, . . . , αk} such that the weight
net is a disjoint union of α-strings of length 2. Hence at least half of the weights not
orthogonal to each αi for i ∈ [1, k] cannot lie in the eigenspace.
We see that the situation is slightly different for even characteristic. Consider the
weight string µ, µ−α, · · · , µ− kα. If we assume that p = 2 and the weight µ lies in Vγ(s)
then µ− 2jα for j > 0 and 2j 6 k cannot lie in Vγ(s). This follows since α(s) = 1 if and
only if 2jα(s) = 1 for j > 0. Thus the lower bound we obtain on the contribution from
the weight string to the codimension of the eigenspace is at least as great for p = 2 as it
is for p 6= 2.
We shall find it convenient to introduce some notation and terminology for the con-
sideration of weights with a given centraliser type. Define a cluster to be a maximal set
of weights such that the difference between any two is a linear combination of roots in
Φs. The set of weight spaces corresponding to the weights in any cluster all lie in the
same eigenspace. For example, consider SL10(K) acting on L(ω3) with centraliser type
X = A3. If we assume that Φs has simple roots {α1, α2, α3} then there is a cluster of size
6 consisting of the following weights:
7 14 11 8 15 12 9 6 3 7 4 11 8 15 12 9 6 3 7 4 1 8 15 12 9 6 3
-3 4 11 8 15 12 9 6 3 -3 4 1 8 15 12 9 6 3 -3 -6 1 8 15 12 9 6 3
Note that we have omitted a factor of 1
10
on each coefficient. For brevity we shall
denote such a cluster by the string · · · 8 15 12 9 6 3 where the dots stand for any integers
which make the expression a weight in Π(ω3). We define a clique to be a collection of
clusters where any two clusters contain weights which differ by a root not in Φs. We shall
present a clique as a list of clusters in a box. Continuing the example above, the following
is a clique:
39
· · · 8 15 12 9 6 3
· · · 8 5 12 9 6 3
· · · 8 5 2 9 6 3
· · · 8 5 2 -1 6 3
· · · 8 5 2 -1 -4 3
· · · 8 5 2 -1 -4 -7
Given a centraliser type X we wish to show that dim
⋃
s∈SX
⋃
γ∈K∗ Vγ(s) < dimV
where SX = {s ∈ Gss | CG(s) of type X}. If s ∈ T then Vγ(s) is a sum of weight spaces
and the set of weights concerned is a union of clusters. By the adjacency principle we see
that at most one cluster in each clique may contribute to the dimension of the eigenspace
Vγ(s) for any s ∈ Gss \ Z and γ ∈ K
∗.
Henceforth, unless it is otherwise clear from the context, we shall fix s ∈ Gss \ Z
and γ ∈ K∗ and so consider a specific eigenspace Vγ(s). By the statement a weight µ
lies in the eigenspace Vγ(s) we mean that the weight space Vµ lies in Vγ(s). Since Π(λ)
denotes the multiset of all weights of a G-module V = L(λ) including multiplicities, we
have |Π(λ)| = dimV . We set
Λ = Λ(s, γ) = {µ ∈ Π(λ) | µ(s) 6= γ}
to be the multiset of weights with multiplicities such that the corresponding weight spaces
do not lie in the eigenspace, so |Λ| = codimVγ(s) and, of course, |Π(λ) \ Λ| = dimVγ(s).
We shall use repeatedly the following strategy. Firstly, we use weight strings to obtain
a lower bound for |Λ|. We then use this value to determine which possibilities, if any, for
X need to be considered further. For each such centraliser type we arrange the weights
into clusters and then cliques to obtain an improved lower bound for dimV − dimVγ(s).
In most cases this will exceed eXλ .
4.3 Criteria for modules
This next proposition will allow us (whenever the conclusion of Premet’s theorem holds)
to list the irreducible modules V for which we cannot immediately conclude that dimE <
dimV , by providing a criterion by which (†) is satisfied. We begin Chapters 5-8 using
this proposition to show in most cases (†) holds and to list the highest weights of modules
40
requiring further consideration. These are provided in tables giving the highest weight λ
of the irreducible module and the rank n of the simple algebraic group concerned.
Recall from Chapter 2 we use the convention that if all roots in Φ have the same length
then we shall regard them as having short length.
Proposition 4.2. Let G be a simple simply connected algebraic group with root system Φ
acting on L(λ). Let µ =
∑n
j=1 ajωj 6 λ be a dominant weight and assume that p > e(Φ).
Set Ψ = 〈αi | ai = 0〉 ⊂ Φ and rΨ = |W (Φ) : W (Ψ)|
|ΦS\ΨS |
2|ΦS |
. Then |Λ(s, γ)| >
∑
rΨ,
where the sum runs over the dominant weights of L(λ). Moreover, if
∑
rΨ > dimG then
(†) holds.
Proof. For any non-central semisimple element s there is at least one root α /∈ Φs. If Φ
consists of roots with different lengths then we may as well assume that α is short. For,
if all short roots of Φ lie in Φs then all long roots would also lie in Φs.
From the definition of the subsystem Ψ we see that α ∈ Ψ if and only if 〈µ, α〉 = 0,
i.e., µ is orthogonal to α. Thus, the roots that lie in ΦS \ ΨS are the short roots not
orthogonal to µ. The stabiliser of µ in W (Φ) is the parabolic subgroup generated by the
reflections in the simple roots αi for which ai = 0; this is preciselyW (Ψ). Hence the Weyl
group orbit W.µ has size |W (Φ) : W (Ψ)|.
We note that for each α′ ∈ ΦS, the number of weights µ
′ ∈ W.µ not orthogonal to
α′ is constant. This follows by observing that a weight µ′ is orthogonal to α′ if and only
if wµ′ is orthogonal to wα′ and W acts transitively on the short roots. The number of
pairs (µ′, α′) ∈ W.µ× ΦS with 〈µ
′, α′〉 6= 0 is |W (Φ) : W (Ψ)| |ΦS \ΨS|. Thus, for a fixed
α ∈ ΦS, the number of weights µ
′ ∈ W.µ such that 〈µ′, α〉 6= 0 is
|W (Φ) : W (Ψ)|
|ΦS \ΨS|
|ΦS|
= 2rΨ.
Henceforth we shall fix a short root α /∈ Φs. Recall that the set of weights of L(λ) is a
union of Weyl group orbits and each Weyl group orbit contains a unique dominant weight
(which we use as an orbit representative). We take the weights of L(λ) and arrange them
in α-strings. Each weight occurs in a string of odd or even length.
41
In an α-string of odd length 2k + 1 there is precisely one weight orthogonal to α; this
weight occurs in position k + 1. The weights in position t and 2k + 2− t both lie in the
same Weyl group orbit of a dominant weight (via wα ∈ W ). On the other hand, in an
α-string of even length 2k there are no weights orthogonal to α. The weights in position t
and 2k+1− t both lie in the same Weyl group orbit of a dominant weight. So weights not
orthogonal to α occur in pairs lying in the same Weyl group orbit of a dominant weight.
As stated above, for a dominant weight µ, the number of weights µ′ ∈ W.µ not
orthogonal to α is 2rΨ. Thus, for every dominant weight, the number of weights of L(λ)
not orthogonal to α is
∑
2rΨ, where the sum runs over the dominant weights of L(λ).
In an even string of length 2k, all 2k weights are counted and in an odd string of length
2k + 1, we see that 2k of the 2k + 1 weights are counted in this calculation.
We now apply the adjacency principle to each α-string to obtain a lower bound for the
codimension of the eigenspace. We shall assume that all weights have multiplicity one.
(If an irreducible module has dominant weights with larger multiplicities then the lower
bound for the codimension of the eigenspace may in fact be greater.) For an α-string of
even length 2k the minimal contribution to the codimension of the eigenspace is k since
there are k pairs of weights adjacent in the α-string. Similarly, for an α-string of odd
length 2k + 1 the minimal contribution to the codimension of the eigenspace is k. We
can arrange all but one of the weights into pairs of weights adjacent in the α-string. The
remaining weight may or may not contribute to the codimension of the eigenspace; we
are unable to conclude either way. Thus, for all α-strings, the minimal contribution to
|Λ(s, γ)| is
∑
rΨ, where the sum runs over the dominant weights of L(λ).
In Chapter 10 we shall consider tensor products of irreducible G-modules in order to
complete the proof of the Main Theorem for modules parameterised by highest weights
that are not p-restricted. In this case we require the next proposition which provides a
criterion which we shall use to show that for all but a short list of modules the conclusion
of the Main Theorem holds. For the remaining modules further study is required to show
that either (†) holds or (♦) is satisfied for all X ∈ I.
42
Proposition 4.3. Let G be a simple simply connected algebraic group and let V = U⊗W
where U and W are irreducible G-modules. Suppose that each eigenspace in U of each
non-central semisimple element has codimension at least d. If d dimW > dimG then (†)
holds.
Proof. Given s ∈ Gss \ Z we decompose U and W into eigenspaces for s, i.e., we may
write U =
⊕
σ Uσ(s) and W =
⊕
τ Wτ (s) where we sum over the eigenvalues σ ∈ K
∗ and
τ ∈ K∗ of s on U and W respectively. Thus we can write the eigenspace of s on V with
eigenvalue ρ in the form
Vρ(s) =
⊕
(σ,τ)
Uσ(s)⊗Wτ (s)
where the sum is over pairs (σ, τ) with στ = ρ.
Fix an eigenvalue ρ ∈ K∗ of s on V . Since by assumption each eigenspace in U has
codimension at least d, for each τ ∈ K∗ we have dimUρτ−1(s) 6 dimU − d and hence
dim (Uρτ−1(s)⊗Wτ (s)) 6 (dimU − d) dimWτ (s).
By summing over τ we obtain
dimVρ(s) 6 (dimU − d) dimW = dimV − d dimW,
i.e., codimVρ(s) > d dimW and the result follows.
In particular, since d > 1, this proposition implies that we need only to consider
modules V where the dimension of both factors U and W is at most that of G.
43
Chapter 5
Groups of type An
We are now ready to begin the proof of the Main Theorem. Throughout this chapter we
shall examine the action of non-central semisimple elements in a simple simply connected
algebraic group G of type Φ = An on an irreducible module V = L(λ) parameterised by
a p-restricted weight λ. We shall obtain a list of irreducible modules for which dimE =
dimV is possible and show that (†) holds for all others. For the remaining modules we
further analyse the weights involved. If we are unable to show (†) holds then we find the
possible root system types of centralisers of non-central semisimple elements and use (♦)
to draw the necessary conclusion. Recall that modules are listed up to duality. Thus the
dual of any module considered should be regarded as also being considered.
We shall prove the following result.
Theorem 5.1. Let G be a simple simply connected algebraic group of type An acting on
an irreducible module V = L(λ) where λ is p-restricted. If dimV 6 dimG+2 then E = V
with the possible exceptions of L(3ω1) for n ∈ [1, 2] with p > 3, L(4ω1) for n = 1 with
p > 3, L(ω1 + ω2) for n = p = 3 and L(ω3) for n = 7; if instead dimV > dimG + 2
then dimE < dimV with the possible exceptions of L(ω3) for n = 8, L(ω4) for n = 7 and
L(2ω2) for n = 3 with p > 2.
This theorem is a consequence of the lemmas which follow in later sections.
44
5.1 Initial survey
Let µ =
∑n
i=1 aiωi 6 λ be a dominant weight. Recall from Section 2.1.1 that we shall
be considering weights up to duality. We begin by using Proposition 4.2 to derive six
conditions (i)-(vi) on the coefficients ai of µ under which (†) holds assuming that n is
sufficiently large. Indeed, once we have shown for a given weight µ that (†) holds then the
same will be true when we add any positive linear combination of fundamental weights.
The minimal contribution to the codimension of an eigenspace for µ′ = µ +
∑n
i=1 ciωi
where ci > 0 for 1 6 i 6 n with subsystem Ψ
′ is at least that for µ with subsystem Ψ as
can be seen by comparing the sizes of the root systems Ψ and Ψ′ and also the orders of
their Weyl groups; the inequality rΨ′ > rΨ holds.
We shall then use these six conditions and Premet’s theorem to determine a short
list of irreducible G-modules requiring further consideration, i.e., those modules for which
more work is required to show that either (†) holds or (♦) is satisfied for all centraliser
types of non-central semisimple elements.
Proposition 5.2. Suppose that µ =
∑n
i=1 aiωi 6 λ is a dominant weight, and that at
least one of the following conditions holds:
(i) n > 6 and aj, ak 6= 0 for some j ∈ [3, n− 3] and k ∈ [j + 2, n];
(ii) n > 5 and aj, ak 6= 0 for some j ∈ [2, n− 3] and k ∈ [j + 2, n− 1];
(iii) n > 10 and ak 6= 0 for some k ∈ [5, n− 4];
(iv) n > 13 and either a4 6= 0 or an−3 6= 0;
(v) n > 7 and a4, an−k 6= 0 for some k ∈ [0, n− 5];
(vi) n > 6 and ak, ak+1 6= 0 for some k ∈ [3, n− 3].
Then (†) holds.
Proof. First assume that both a3 6= 0 and an 6= 0; then the subsystem Ψ = 〈αi | ai = 0〉
is contained in Φ(A2An−4). We find that |W (Φ) : W (Ψ)| >
1
6
(n+ 1)n(n− 1)(n− 2) and
45
|Φ \ Ψ| > 2(4n − 9). Therefore rΨ > rA2An−4 =
1
6
(n − 1)(n − 2)(4n − 9) which exceeds
dimG = n(n+2) for n > 6 and (†) is satisfied. If we instead assume that both aj 6= 0 and
ak 6= 0 for 3 6 j < k 6 n, k > j + 1 and j 6= n− 2 then Ψ ⊂ Φ(Aj−1Ak−j−1An−k). Since
rAj−1Ak−j−1An−k > rA2An−4 for such values of j and k we see that condition (i) ensures that
(†) holds.
We treat condition (v) similarly. If both a4 6= 0 and an 6= 0 then Ψ ⊂ Φ(A3An−5) and
so rΨ > rA3An−5 =
1
24
(n− 1)(n− 2)(n− 3)(5n− 16); this exceeds dimG for n > 7. If we
assume that both a4 6= 0 and an−k 6= 0 for k ∈ [0, n − 5] then Ψ ⊂ Φ(A3An−5−kAk) and
since rA3An−5−kAk > rA3An−5 for k ∈ [0, n− 5] we see that (†) holds.
Next suppose that both a2 6= 0 and an−1 6= 0. Then Ψ ⊂ Φ(A1An−4A1) and so
rΨ > rA1An−4A1 = (n− 1)(n− 2)
2 which exceeds dimG for n > 5. If we now assume that
both aj 6= 0 and ak 6= 0 where 1 < j < k < n and k > j+1 then Ψ ⊂ Φ(Aj−1Ak−j−1An−k)
and we can use the fact that rAj−1Ak−j−1An−k > rA1An−4A1 to conclude that (†) holds here
for n > 5. Thus condition (ii) ensures that (†) holds.
Assume that ak 6= 0 for k ∈ [4, n − 3] so that Ψ ⊂ Φ(Ak−1An−k) and we find that
rΨ > rAk−1An−k =
(
n−1
k−1
)
. If we take k = 5 then we see rΨ > dimG for n > 10. For
k ∈ [5, n− 4] since rAk−1An−k > rA4An−5 we see that (†) holds and as such we cannot have
ak 6= 0. If k = 4 we have rΨ > dimG for n > 13. Thus conditions (iii) and (iv) both
ensure that (†) holds.
Finally we treat condition (vi); assume that both ak 6= 0 and ak+1 6= 0 for k ∈ [3, n−3].
Then we have Ψ ⊂ Φ(Ak−1An−k−1) and rΨ > rAk−1An−k−1 =
(n−1)!
k!(n−k)!
(nk − k2 + n) which
exceeds dimG for n > 6. This completes the proof of the proposition.
We shall split the initial analysis of weights into three sections according as n > 10,
n ∈ [5, 9] or n ∈ [1, 4].
The modules that have dimension at most dimG + 2 are the following: L(ω1) for
n > 1, L(3ω1) for n ∈ [1, 2] with p > 3, L(4ω1) for n = 1 with p > 3, L(ω1 + ω2) for
n = p = 3, L(ω2) and L(2ω1) (p > 2) for all n, the module L(ω3) for n ∈ [5, 7], and
the adjoint module L(2ω1) for n = 1 with p > 2 and L(ω1 + ωn) for n > 2. Recall from
46
Section 3.1 that we need only consider modules with dimension greater than or equal to
dimB, where B is a Borel subgroup of G and those that are not adjoint modules. In type
An, we have dimB = |Φ
+(G)|+ rankT = 1
2
n(n+ 3) in which case, for n > 2, the natural
module L(ω1) with dimension n+ 1 and L(ω2) with dimension
1
2
n(n+ 1) are too small.
Lemma 5.3. Suppose that n > 10. If dimV > dimG + 2 then (†) holds except possibly
for the modules with highest weight 3ω1 (p > 3), ω1+ω2, ω3, ω2+ωn and 2ω1+ωn (p > 2)
for n > 10 and ω4, ω1+ω3, 2ω2 (p > 2), 2ω1+ω2 (p > 2) and 4ω1 (p > 3) for n ∈ [10, 12].
Proof. Assume that n > 13. By (iv) of Proposition 5.2 any dominant weight µ 6 λ is of
the form µ = µ1 + µ2 where µ1 =
∑3
i=1 aiωi and µ2 =
∑n
i=n−2 aiωi. Set m1 =
∑3
i=1 iai
and m2 =
∑3
i=1 ian+1−i; we shall assume that m1 > m2 since we are considering modules
up to duality. From [13, p.69] we can see that µ1 =
∑3
i=1 aiωi >
∑3
i=1(ai − bi)ωi + ωj
where 0 6 bi 6 ai and j =
∑3
i=1 ibi. The expression
∑3
i=1 biωi − ωj can be written as
a non-negative linear combination of simple roots; the coefficient of each αk is clearly
positive for all k ∈ [1, 3], is (j−k)(n+1) for k ∈ [4, j−1] and is zero otherwise. If m1 > 4
then we can take bi, i = 1, 2, 3 such that 4 6 j 6 6. Thus
µ′ = µ−
(
3∑
i=1
biωi − ωj
)
< µ 6 λ
is a dominant weight less than the highest weight and if we write µ′ =
∑n
i=1 a
′
iωi then
a′k 6= 0 for some k ∈ [4, 6]. Thus (†) is satisfied by condition (iv), so we may assume that
m1 6 3 and m2 6 3.
Suppose that m1 = 3 so µ1 is one of ω3, ω1 + ω2 or 3ω1. If a3 6= 0 then µ1 = ω3 and,
by condition (i) of Proposition 5.2, we must have m2 = 0 for n > 6. In any case since
3ω1 > ω1 + ω2 > ω3 we can appeal to Premet’s theorem to conclude that m2 = 0. Thus
the three weights ω3, ω1 + ω2 and 3ω1 require further consideration.
Next suppose that m1 = 2 so µ1 is either ω2 or 2ω1. Assume that µ1 = ω2 and m2 = 2,
so either µ2 = ωn−1 or 2ωn. If µ2 = ωn−1 then by condition (ii) of Proposition 5.2 we
see that (†) holds for n > 5 and if µ2 = 2ωn then (†) holds again by condition (ii) since
47
ωn−1 < 2ωn. If µ1 = 2ω1 then again (†) holds when m2 = 2 since ω2 < 2ω1. Hence m2 6 1
and the weights that require further consideration are ω2 + ωn and 2ω1 + ωn since, as we
have already mentioned, the modules L(ω2) and L(2ω1) have dimension no larger than
dimG.
Lastly, if m1 = 1 then m2 6 1 in which case the possible weights are ω1 which has
dimension less than dimB and ω1 + ωn which is the adjoint module.
Now assume that 10 6 n 6 12. As above we see from condition (iii) of Proposition
5.2 that any dominant weight µ 6 λ is of the form µ = µ1+µ2 where µ1 =
∑4
i=1 aiωi and
µ2 =
∑n
i=n−3 aiωi. Set M1 =
∑4
i=1 iai and M2 =
∑4
i=1 ian+1−i. In the same way as before
if M1 > 4 we can take bi for i ∈ [1, 4] with 0 6 bi 6 ai and 5 6 j 6 8 where j =
∑4
i=1 ibi.
Thus
µ′ = µ−
(
4∑
i=1
biωi − ωj
)
< µ 6 λ
is a dominant weight less than the highest weight with a′k 6= 0 for k ∈ [5, 8] where
µ′ =
∑n
i=1 a
′
iωi. We need to look further at the cases n = 11 with j = 8 and n = 10
with j ∈ [7, 8] since condition (iii) of Proposition 5.2 cannot be used here. Suppose that
n = 11. The only weight for which j = 8 and a′k = 0 for k ∈ [5, 7] is 2ω4. Here ω8 < 2ω4
and ai = bi for i ∈ [1, 4]. Similarly for n = 10 the only weights for which j ∈ [7, 8] and
a′k = 0 for k ∈ [5, 6] are 2ω4 and ω3 + ω4; however we do not need to consider the latter
weight by condition (vi) of Proposition 5.2. Indeed, since 2ω4 > ω3 + ω5 we can use (ii)
to conclude that (†) holds for 2ω4. Thus we can assume that M1 6 4 and, by considering
modules up to duality, M2 6 4.
If M1 = 4 then µ is one of ω4, ω1 + ω3, 2ω2, 2ω1 + ω2 or 4ω1. We may assume
that M2 = 0 by condition (v) of Proposition 5.2 and Premet’s theorem since if ν ∈
{ω1+ω3, 2ω2, 2ω1+ω2, 4ω1} then ω4 < ν. Thus the weights requiring further consideration
are ω4, ω1 + ω3, 2ω2, 2ω1 + ω2 and 4ω1. Similarly if M1 = 3 then we may assume that
M2 = 0 by (i) and Premet’s theorem. The weights requiring further consideration here
are ω3, ω1 + ω2 and 3ω1. If M1 = 2 then (ii) implies that M2 6 1. Therefore it remains
to consider the weights ω2 + ωn and 2ω1 + ωn.
48
Lemma 5.4. Suppose that dimV > dimG+2. If 5 6 n 6 9 then (†) holds except possibly
for the modules with highest weights ω1 + ω3, 2ω1 + ω3 (p > 2), ω2 + ω3, 2ω1 (p > 2), 3ω1
(p > 3), 4ω1 (p > 3), 2ω2 (p > 2), ω1 + ω2, 2ω1 + ω2 (p > 2), 3ω1 + ω2 (p > 3), 2ω1 + ωn
(p > 2) and ω2 + ωn for n ∈ [5, 9], ω1 + 2ω2 (p > 2) and 5ω1 (p > 5) for n = 5, ω3 for
n ∈ [8, 9], ω4 for n ∈ [7, 9] and ω5 for n = 9.
Proof. Suppose that a3 6= 0 and n > 6. By condition (i) of Proposition 5.2 we can
assume that ak = 0 for k ∈ [5, n] and by condition (vi) of Proposition 5.2 we may
assume that a4 = 0. We can take a3 6 1 by condition (ii) of Proposition 5.2 since
2ω3 > ω2 + ω4. Similarly, we can assume by (ii) that ai 6= 0 for at most two i ∈ [1, 3] as
ω1+ω2+ω3 > ω2+ω4. If a1 6= 0 and a2 = 0 we have a1 6 2 by (ii) since 3ω1+ω3 > ω2+ω4
and if a1 = 0 and a2 6= 0 we have a2 = 1 since 2ω2 + ω3 > ω1 + 2ω3. Thus for n > 6 we
must consider further the weights ω1 + ω3, 2ω1 + ω3 and ω2 + ω3.
Suppose that a3 6= 0 and n = 5. We can only use condition (ii) of Proposition 5.2 in
this case. If at least three coefficients of µ have non-zero coefficients then Ψ is contained
in either Φ(A2) or Φ(A
2
1). In the former case we see by (ii) that (†) holds for ω1+ω2+ω3
since ω2 + ω4 < ω1 + ω2 + ω3. Similarly for the weight ω3 + ω4 + ω5. In the latter case
(†) holds using the calculation in (ii) of Proposition 5.2 as rA2
1
> rA3
1
. Thus we can
assume that at most two coefficients of µ are non-zero. We can take a3 = 1 by (ii) since
2ω3 > ω2+ω4. If a1 6= 0 we can take a1 6 2 by (ii) since ω3+3ω1 > 2ω3 and if a2 6= 0 we
can take a2 = 1 since 2ω2 + ω3 > ω1 + 2ω3. Therefore for n = 5 we must consider further
the weights ω1 + ω3, 2ω1 + ω3 and ω2 + ω3.
Suppose that a4 6= 0 and n > 7 (otherwise we are in an earlier case). We may assume
that a4 = 1 by (ii) since 2ω4 > ω3+ω5. We may also assume that ak = 0 for k = 3 by (vi),
for k = 2 by (ii) and k ∈ [5, n] by (v) of Proposition 5.2. If a1 6= 0 then Ψ ⊂ Φ(A2An−4)
and as in (i) we find that rΨ > dimG. We note therefore that (†) certainly holds if a4 6= 0
and any other coefficient of µ is non-zero. Hence we need to consider further the weight
ω4 for n ∈ [7, 9].
Suppose that a5 6= 0 and n = 9. We may assume that a5 = 1 by (ii) since 2ω5 > ω4+ω6.
We may assume that ak = 0 for k ∈ [2, 3] by (ii), for k = 4 by (vi) and for k = 1 by
49
an argument as in (v) since in this case we have Ψ ⊂ Φ(A3A4). By duality we may also
assume that ak = 0 for k ∈ [6, 9]. Hence we need to consider the weight ω5 for n = 9
further.
Now suppose that a2 6= 0. Assume ak 6= 0 for some k 6= 2. We have dealt with the
cases ak 6= 0 for k ∈ [3, 5] above. Indeed, we may assume by condition (ii) of Proposition
5.2 that if a2 6= 0 then ak = 0 for k ∈ [4, n− 1].
If a1 > 2, a2 = 0 and ak 6= 0 for any k ∈ [3, n − 1] then (†) holds by the previous
paragraph as 2ω1 > ω2. Thus we may assume that a1 = 1 if a2 = 0 and ak 6= 0 for
k ∈ [3, n− 1] whence, up to duality, we are in an earlier case.
Thus we are left to consider weights of the form µ = a1ω1 + a2ω2 + anωn. We quickly
see that an 6 1 by (ii) and Premet’s theorem since ωn−1 < 2ωn. We shall treat the cases
an = 0 and an = 1 separately.
Case I: an = 0. If both a1 6= 0 and a2 6= 0 then we may assume that a1 6 3, a2 6 3
and a1+a2 6 4 by (ii) since 2ω1+2ω2 > ω2+ω4; thus we may assume that we do not have
a1 = a2 = 2. From the analysis above we see that (†) holds for ω1 + ω2 + ω4 so the same
is true for ω1+3ω2 > ω1+ω2+ω4. Note that by the calculation in (i) of Proposition 5.2,
we need to consider further the weight ω1 + 2ω2 only for n = 5 since ω1 + 2ω2 > ω1 + ω4.
The weights ω1 + ω2, 2ω1 + ω2 and 3ω1 + ω2 require further consideration.
If a1 6= 0 and a2 = 0 we may assume that a1 6 5 by (ii) since 6ω1 > ω2+ω4. We need
only consider 5ω1 for n = 5 by the calculation in (i) since 5ω1 > ω1 + ω4. The weights
3ω1 and 4ω1 require further consideration for n ∈ [5, 9].
If a1 = 0 and a2 6= 0 we may assume that a2 6 2 by (ii) since 3ω2 > ω2+ω4. However
the modules L(ω2) and L(2ω1) both have dimension less than dimG.
Case II: an = 1. Suppose that an = 1. Then we can assume that at most one of a1
and a2 is non-zero for n > 6 by (i) since ω1 + ω2 + ωn > ω3 + ωn and for n = 5 we find
that rA2 = 48 > 35 = dimG. If µ = a1ω1+ωn then a1 6 2 since 3ω1+ωn > ω1+ω2+ωn.
If µ = a2ω2 + ωn we can assume that a2 = 1 by the calculation in (ii) since 2ω2 + ωn >
ω1+ω3+ωn. Hence further work is needed for the modules with highest weights ω2+ωn
and 2ω1 + ωn.
50
We shall now consider the low rank cases. The modules outstanding from the following
lemma will be considered later in Section 5.5.
Lemma 5.5. Suppose that 1 6 n 6 4. If dimV > dimG+2 then (†) holds except possibly
for the modules with highest weights 5ω1 and 6ω1 for n = 1 with p > 5, 3ω1 for n ∈ [3, 4]
with p > 3, 4ω1 for n ∈ [2, 4] with p > 3, 2ω1 + ω2 for n ∈ [2, 4] with p > 2, 2ω2 (p > 2),
ω1 + ω2 and 2ω1 + ωn (p > 2) for n ∈ [3, 4], ω2 + ω3 and ω1 + ω3 for n = 4, and 3ω1 + ω2
(p > 3) and 2ω1 + 2ω2 (p > 2) for n = 2.
Proof. We calculate the value rΨ for all possible Ψ ⊂ Φ and use Premet’s theorem to
show that (†) holds for all but a small number of weights.
Case I: n = 4. We have dimG = 24 and rΨ = 60, 27, 12, 7, 3 or 1 according as
Ψ = ∅, A1, A21, A2, A2A1 or A3. Let µ =
∑4
i=1 aiωi 6 λ be a dominant weight. Since
rA1 > dimG we see that (†) holds if we have ai 6= 0 for more than two i ∈ [1, 4] and
by Premet’s theorem (†) holds for any weight µ′ < µ with more than two coefficients
non-zero.
First let us assume that a3 = a4 = 0. If a1 = 0 also then we may assume that a2 6 2
due to the fact that ω1 + ω2 + ω3 < 3ω2. Thus we must consider further the weight 2ω2.
Similarly if a2 = 0 then a1 6 4; we have
5ω1 > 3ω1 + ω2 > ω1 + 2ω2 > 2ω1 + ω3 > ω2 + ω3
so we find by Premet’s theorem that |Λ| > 1 + 7 + 7 + 12 + 12 = 39. Thus the weights
3ω1 and 4ω1 require further consideration. If both a1 and a2 are non-zero then we may
assume that a1 6 2 and a2 6 1 since (†) is satisfied for both ω1 + 2ω2 and 3ω1 + ω2 as
can be seen from the calculation for 5ω1 above. Thus the weights ω1 + ω2 and 2ω1 + ω2
require further consideration.
Now let us assume that µ = apωp+ aqωq where p ∈ {1, 2}, q ∈ {3, 4} and ap > ar > 0.
Let us suppose first that q = 3. If p = 2 we may assume that a2 6 1 from the remark
above since 2ω2 + ω3 > ω1 + ω2 + ω4. If p = 1 we may assume that a1 6 1 since
2ω1 + ω3 > ω2 + ω3 > ω1 + ω4 so that |Λ| > 12 + 12 + 7 = 31, hence (†) holds. Now
51
suppose that q = 4. If p = 2 then we may assume that a2 6 1 since 2ω2+ω4 > ω1+ω3+ω4.
If p = 1 then we may assume that a1 + a4 6 3; we have 3ω1 + ω4 > ω1 + ω2 + ω4 and
since 2ω1 + 2ω4 > 2ω1 + ω3 we can use Premet’s theorem and the calculation above for
2ω1+ω3. Recall that we do not consider the modules L(ω1), L(ω2) and L(2ω1) since they
have dimension smaller than dimG. Note that we shall concern ourselves with L(ω1+ω3)
rather than L(ω2 + ω4) as we are considering modules up to duality. Thus the weights
requiring further consideration are ω1 + ω3, ω2 + ω3 and 2ω1 + ω4.
Case II: n = 3. We have dimG = 15 and rΨ = 12, 5, 2, or 1 according as Ψ = ∅, A1, A21
or A2. Suppose first that a2 = a3 = 0. Since ν < 5ω1 for ν ∈ {3ω1+ω2, ω1+2ω2, 2ω1+ω3}
we may assume that a1 6 4. Thus further consideration is required for the weights 3ω1
and 4ω1. Suppose that µ = a1ω1 + a3ω3 with a1 > a3 > 0. Since ω1 + ω2 + ω3 < 3ω1 + ω3
and ν < 2ω1 + 2ω3 for ν ∈ {ω2 + 2ω3, 2ω1 + ω2, 2ω2} we can assume that a1 6 2 and
a3 = 1. Thus we need to consider further only the weight 2ω1+ω3 since the module with
highest weight ω1 + ω3 is the adjoint module.
Now suppose that µ =
∑3
i=1 aiωi with a2 6= 0. We may assume that a3 = 0 as (†)
is satisfied if ai 6= 0 for each i ∈ [1, 3]. This is because ν < ω1 + ω2 + ω3 for each
ν ∈ {2ω3, 2ω1, ω2}. Consequently we may assume that a2 6 2 since ω1+ω2+ω3 < 3ω2. If
a2 = 2 then a1 = 0 since ν < ω1+2ω2 for each ν ∈ {2ω1+ω3, ω2+ω3, ω1}. If a2 = 1 then
a1 6 2 since ω1 + 2ω2 < 3ω1 + ω2 and we have just observed that (†) holds for ω1 + 2ω2.
Thus we are left to consider further the weights ω1 + ω2, 2ω1 + ω2 and 2ω2.
Case III: n = 2. Here dimG = 8 and we find that r∅ = 3 and rA1 = 1. If µ = a1ω1
then we may assume that a1 6 4 since ν < 5ω1 for each ν ∈ {3ω1+ω2, ω1+2ω2, 2ω1, ω2}.
Thus the weights 3ω1 and 4ω1 require further consideration. Suppose that µ = a1ω1+a2ω2
with a1 > a2 > 0. Then we have a1 6 3 and a2 6 2 with a1 + a2 < 5 since ν < 3ω1 + 2ω2
for each ν ∈ {ω1 + 3ω2, 2ω1 + ω2} and η < 4ω1 + ω2 for each η ∈ {2ω1 + 2ω2, ω1 + ω2}.
We shall need to consider further the weights 2ω1 + ω2, 3ω1 + ω2 and 2ω1 + 2ω2.
Case IV: n = 1. We have dimG = 3; here of course r∅ = 1. By listing the eight
weights for L(7ω1) we see, using the adjacency principle, for any γ ∈ K
∗ and s ∈ Gss \ Z
that |Λ| > 4. Thus we need to further investigate the weights kω1 for k ∈ [5, 6] since
52
L(kω1) has dimension k + 1.
5.2 Weight string analysis
From the analysis in the previous sections, the remaining p-restricted highest weights λ
for n > 1 appear in Table 5.1. We reference the lemma in which each module is treated
and the conclusion drawn. Although it is not stated explicitly in the table we shall assume
that the characteristic is such that each weight is p-restricted. Thus, for example, we list
3ω1 for n ∈ [3,∞) under the assumption that p > 3. If µ =
∑n
i=1 aiωi then we shall say
that µ has level j where j =
∑n−1
i=1 iai + an. We shall order the weights in the table and
organise the lemmas in the next section according to levels of weights.
It is worthwhile to remark that the dimension of each module above depends on
characteristic; this is not clear from Table 5.1.
We shall now begin a sequence of lemmas working through the possible modules. In
each of the succeeding lemmas, we take s ∈ Gss \ Z and γ ∈ K
∗. As detailed in Section
4.1, we aim to show that (†) is satisfied for each s ∈ Gss \Z or else that (♦) holds for each
t ∈ SX ∩ T and X ∈ I. First we consider irreducible modules parameterised by weights
with level 5.
Lemma 5.6. Let G act on an irreducible module V where we take V to be one of L(ω2+
ω3), L(2ω1 + ω3) (p > 2) and L(3ω1 + ω2) (p > 3) for n ∈ [5, 9] and L(ω1 + 2ω2) (p > 2)
and L(5ω1) (p > 5) for n = 5. Then, in each case, (†) holds.
Proof. In this lemma, the weights concerned are related by the partial ordering
5ω1 > 3ω1 + ω2 > ω1 + 2ω2 > 2ω1 + ω3 > ω2 + ω3 > ω1 + ω4 > ω5.
Consider the module with highest weight ω2+ω3. We find for this module that there are at
least rA1An−3 =
1
2
(n−1)(3n−4) weights not in the eigenspace Vγ(s). Since ω1+ω4 < ω2+ω3,
by Premet’s theorem 2.6 the weights in the Weyl group orbitW.(ω1+ω4) occur as weights
of L(ω2 + ω3) so we may include their contribution rA2An−4 =
1
6
(n − 1)(n − 2)(4n − 9)
53
λ n Lemma λ n Lemma
ω2 + ωn [8,∞) 5.8 (†) 2ω1 + ω2 [5, 12] 5.7 (†)
[5, 7] 5.8 (♦) 4 5.16 (†)
2ω1 + ωn [5,∞) 5.8 (†) 3 5.18 (†)
4 5.17 (†) 2 5.23 (♦)
3 5.18 (♦) 4ω1 [5, 12] 5.7 (†)
ω3 [14,∞) 5.12 (♦) 4 5.16 (♦)
[9, 13] 5.13 (♦) 3 5.18 (♦)
8 5.13 (♦)a 2 5.23 (♦)
ω1 + ω2 [5,∞) 5.14 (♦) ω5 9 5.9 (†)
4 5.17 (♦) ω2 + ω3 [5, 9] 5.6 (†)
3 (p 6= 3) 5.21 (♦) 4 (p 6= 3) 5.15 (†)
3ω1 [5,∞) 5.14 (♦) 4 (p = 3) 5.15 (♦)
4 5.17 (♦) 2ω1 + ω3 [5, 9] 5.6 (†)
3 5.19 (♦) ω1 + 2ω2 5 5.6 (†)
ω4 [11, 12] 5.9 (†) 3ω1 + ω2 [5, 9] 5.6 (†)
[9, 10] 5.9 (♦) 2 (p 6= 3, 5) 5.22 (†)
8 5.10 (♦) 2 (p = 5) 5.22 (♦)
7 5.11 (♦)b 5ω1 5 5.6 (†)
ω1 + ω3 [5, 12] 5.7 (†) 1 5.24 (♦)
4 5.16 (♦) 2ω1 + 2ω2 2 (p 6= 2, 5) 5.22 (†)
2ω2 [6, 12] 5.7 (†) 2 (p = 5) 5.22 (♦)
5 (p 6= 2, 3) 5.7 (†) 6ω1 1 5.24 (♦)
5 (p = 3) 5.7 (♦)
4 5.16 (♦)
3 5.20 (♦)c
Except for: aX = A32 and A4A3,
bX = A23,
cX = A21.
Table 5.1: Possible weights in type An, n > 1
to |Λ|. Thus there are at least rA2An−4 + rA1An−3 weights not in the eigenspace, which is
greater than dimG for n ∈ [5, 9], as required. The other modules are each above ω2 + ω3
in the partial ordering. Hence, by using Premet’s theorem, (†) holds for these also for
n ∈ [5, 9].
In the next lemma we deal with weights with level 4. We find here that simply counting
weight strings is insufficient to conclude that |Λ| is large enough to satisfy (†). We employ
the information [18] on multiplicities of weights to assist our analysis.
Lemma 5.7. Let G act on the irreducible module V where we take V for n ∈ [5, 12] to
54
be one of L(ω1 + ω3), L(2ω2) (p > 2), L(2ω1 + ω2) (p > 2) and L(4ω1) (p > 3). Then in
each case (†) holds unless n = 5 and p = 3 for L(2ω2) in which case (♦) is satisfied for
all X ∈ I.
Proof. We are considering the four irreducible modules with highest weights satisfying
the partial ordering
4ω1 > 2ω1 + ω2 > 2ω2 > ω1 + ω3 > ω4.
We wish to show that for each non-central s ∈ Gss and γ ∈ K
∗ the condition (†) holds in
each case. By similar calculations to those above, the module with highest weight ω1+ω3
for n ∈ [5, 12] has at least rA1An−3 =
1
2
(n − 1)(3n − 4) weights µ ∈ W.(ω1 + ω3) in |Λ|.
Indeed, ω4 < ω1+ω3, so we can employ Premet’s theorem to add an extra rA3An−4 =
(
n−1
3
)
weights from W.ω4 to |Λ|. We see that (†) is satisfied unless n ∈ [5, 6]; we shall return to
these two cases shortly.
Take α /∈ Φs; the module with highest weight 2ω2 for n ∈ [5, 12] has at least n − 1
weights µ ∈ W.(2ω2) in weight strings of the form µ, µ−α, µ−2α, hence at least rA1An−2 =
n−1 weights in Λ. Using Premet’s theorem to include weight strings from the Weyl group
orbits W.ω4 and W.(ω1 + ω3) we are done unless n = 5 since |Λ| >
(
n−1
3
)
+ 1
2
(n− 1)(3n−
4) + (n− 1) satisfies (†). If n = 5 then |Λ| > 4 + 22 + 4 = 30, hence the module L(2ω2)
requires further consideration in this case.
For the highest weight 2ω1+ω2, we need only consider the case n = 5 from the previous
paragraph and Premet’s theorem. By adding together the various values of rΨ as we run
through the subsystems Ψ = A3, A3A1, A2A1 and A4 corresponding to the four dominant
weights 2ω1+ω2, 2ω2, ω1+ω3 and ω4 we find that |Λ| > 9+4+22+4 = 39 > 35 = dimG.
For 4ω1 we are done by the previous calculations using Premet’s theorem 2.6.
It remains for us to consider the modules L(ω1 + ω3) for n ∈ [5, 6] and L(2ω2) for
n = 5.
First we turn to the module with highest weight ω1 + ω3 for n = 5; it has dimension
90 for characteristic 2 and 105 otherwise.
In Figure 5.1 on the left we tabulate data about the weights from Lu¨beck’s tables.
55
In the first column we index the Weyl group orbits of dominant weights listed in the
second column; the third column gives the size of each of these orbits. The last columns
give the multiplicities mω of each weight in the orbit depending on the characteristic of
K. On the right we tabulate the weight strings that occur and a lower bound for their
contribution to the codimension of the eigenspace. We denote by l the minimal possible
contribution to the codimension from each weight string for the possible characteristics.
The various contributions are summed and a lower bound for |Λ| is provided in the last
row and column. We shall provide further explanation shortly. A weight labelled µi is
one which lies in the Weyl group orbit of the weight indexed by i.
i ω |W.ω| mω
p 6= 2 p = 2
2 ω1 + ω3 60 1 1
1 ω4 15 3 2
Weight No. of l
strings strings p 6= 2 p = 2
µ2 16
µ2 µ2 16 16 16
µ2 µ1 µ2 6 12 12
µ1 1
µ1 µ1 4 12 8
Lower bound on |Λ| 40 36
Figure 5.1: (λ, n) = (ω1 + ω3, 5)
We give full details of the weight string calculations. We calculate |W (A1A2)| = 12,
|W (A3A1)| = 48 and |W (A5)| = 720. For each µ ∈W.(ω1+ω3), there are 8, 8 and 3 roots
α ∈ Φ(A5) such that 〈µ, α〉 = 0, 1 and 2. Similarly there are 14 or 8 roots α ∈ Φ(A5)
according as 〈µ1, α〉 = 0 or 1. Fix some α ∈ Φ(A5). We see that there are
720
12
.8
30
= 16
weights µ ∈ W.(ω1 + ω3) with α-string µ, 16 weights with α-string µ, µ− α and
720
12
.3
30
= 6
weights with α-string µ, µ − α, µ − 2α; these strings are of the form µ2 in the first case,
µ2, µ2 in the second and µ2, µ1, µ2 in the third. There are
720
48
.14
30
= 7 weights µ1 ∈ W.ω4
with 〈µ1, α〉 = 0; six of these occur in the α-strings of length 3 above, leaving just one
with α-string simply µ1. There are
720
48
.8
30
= 4 weights µ1 ∈ W.ω4 with α-string µ1, µ1.
We may assume that α /∈ Φs. There are 16 strings of the form µ2, µ2, i.e., there
are 16 weight strings consisting of two weights both in the Weyl group orbit of ω1 + ω3
which differ by α. Given such a string, the adjacency principle states that the weight
spaces corresponding to both weights cannot both lie in the eigenspace by the choice of
α. Thus for each weight string µ2, µ2 at least one of the two weight spaces cannot lie
56
in the eigenspace and there is a contribution to |Λ| of at least dimVµ2 = mµ2 = 1 for
all characteristics. There are 16 such strings, so the contribution l to |Λ| is at least 16.
Similarly for each of the 4 weight strings of the form µ1, µ1 at least one of the weight
spaces corresponding to these weights cannot lie in the eigenspace. Since dimVµ1 = 3 or
2 when p 6= 2 or p = 2 the minimal contribution to |Λ| from such weight strings is either
12 if p 6= 2 or 8 if p = 2.
There are 6 weight strings of the form µ2, µ1, µ2, i.e., there are 6 weight strings con-
sisting of three weights µ−2α µ−α µ where the first and third weights lie in W.(ω1+ω3)
and the second lies in W.ω4. We wish to find the minimal contribution of each such
weight string to |Λ|. For a given weight string we cannot have two weight spaces corre-
sponding to two adjacent weights in the root string (i.e., those differing by a root outside
Φs) both lying in the eigenspace. In order to obtain the maximum possible dimension
of the eigenspace (and so the minimal possible codimension of the eigenspace), for each
weight string we must either have the weight spaces corresponding to both µ− 2α and µ
contained in the eigenspace or else the weight space corresponding to the weight µ − α
in W.ω4 contained in the eigenspace. When p 6= 2 the minimal contribution from the 6
weight strings to |Λ| is 18 in the first case and 12 in the second case; clearly the minimal
contribution occurs in the second case. Suppose that p = 2. If the weight µ − 2α lies
in the eigenspace then the weights µ− α and µ both cannot lie in the eigenspace by the
adjacency principle and the fact that 2α(s) = 1 if and only if α(s) = 1. Thus, for each
weight string of length three, precisely one of the weights µ − 2α, µ − α or µ lies in the
eigenspace. By taking these possibilities in turn, we see that the contribution from the
6 weight strings to |Λ| is 18, 12 or 18, repectively. Thus the minimal contribution to |Λ|
occurs when the weight µ− α in each of the 6 weight strings lies in the eigenspace.
The weight spaces with weight string of the form µ2 may or may not lie in the
eigenspace. So we cannot draw a conclusion regarding the contribution to |Λ|. The
same holds for the weight strings of the form µ1.
We present this information in Figure 5.1 and adding together the minimal contribu-
tions l from the various weight strings we find that we have |Λ| > 36 or |Λ| > 40 according
57
as p = 2 or p 6= 2. In either case we have |Λ| > dimG.
We carry out analogous calculations for the module with highest weight ω1 + ω3 for
n = 6. We see in Figure 5.2 that |Λ| > dimG for all p, so in this case we are done.
i ω |W.ω| mω
p 6= 2 p = 2
2 ω1 + ω3 105 1 1
1 ω4 35 3 2
Weight No. of l
strings strings p 6= 2 p = 2
µ2 35
µ2 µ2 25 25 25
µ2 µ1 µ2 10 20 20
µ1 5
µ1 µ1 10 30 20
Lower bound on |Λ| 75 65
Figure 5.2: (λ, n) = (ω1 + ω3, 6)
Now we consider 2ω2 for n = 5. Note that we do not consider p = 2 here since we
are assuming that highest weights are p-restricted. We perform similar calculations to
those above and display the results in the right-hand table in Figure 5.3. We find that
|Λ| > dimG except in characteristic 3.
i ω |W.ω| mω
p 6= 2, 3 p = 3
3 2ω2 15 1 1
2 ω1 + ω3 60 1 1
1 ω4 15 2 1
Weight No. of l
strings strings p 6= 2, 3 p = 3
µ3 7
µ3 µ2 µ3 4 4 4
µ2 12
µ2 µ2 16 16 16
µ2 µ1 µ2 6 12 6
µ1 1
µ1 µ1 4 8 4
Lower bound on |Λ| 40 30
Figure 5.3: (λ, n) = (2ω2, 5)
From the fourth and fifth columns in the left-hand table in Figure 5.3 we note that
when p 6= 2, 3 the added information that mω4 = 2 allows us to conclude that |Λ| > 40;
yet we must argue further for p = 3 when mω4 = 1.
If p = 3, we show in Figure 5.3 that (♦) is satisfied unless X = ∅ since e∅2ω2 = 30. If
X = ∅ then, by listing the 90 weights for this module, we can arrange some of these into
cliques (as given below) in order to show that (†) is satisfied. Note that we have omitted
a factor of 1
3
on each coefficient.
4 2 6 4 2 1 5 6 4 2 4 8 6 4 2 1 2 3 4 2 -2 2 6 4 2
4 2 3 4 2 1 5 3 4 2 4 5 6 4 2 1 -1 3 4 2 -2 2 3 4 2
4 2 3 1 2 1 5 3 1 2 4 5 3 4 2 1 -1 0 4 2 -2 2 3 1 2
4 2 3 1 -1 1 5 3 1 -1 4 5 3 1 2 1 -1 0 1 2 -2 2 3 1 -1
4 5 3 1 -1
58
By taking these cliques and the corresponding cliques obtained by reversing the order
of the coefficients and changing signs, we see that d∅2ω2 > 2(4+4.3) = 32 > 30 = e
∅
2ω2
.
The next lemma differs from previous ones as the highest weights of the irreducible
modules concerned depend on the rank n. As usual we calculate the minimum contribution
to the codimension of the eigenspace for each weight string. If some of the weights have
multiplicity greater than one (as indicated in Lu¨beck’s tables for n 6 20), then the
corresponding contribution may in fact be greater.
Lemma 5.8. Let G act on one of the irreducible modules L(ω2 + ωn) and L(2ω1 + ωn)
(p > 2) both for n ∈ [5,∞). Then (†) holds for both modules, unless n ∈ [5, 7] for
L(ω2 + ωn), in which case (♦) is satisfied for all X ∈ I.
Proof. Consider the module with highest weight ω2 + ωn for general n. As displayed in
the bottom table in Figure 5.4, we have |Λ| > 1
2
(3n2 − 7n + 6), (note that by Premet’s
theorem we know that each weight in W.ω1 has multiplicity at least one). This is larger
than dimG for n > 11. Thus we need only consider the possibilities n ∈ [5, 10] and we
may use the information about multiplicities in the tables to calculate better lower bounds
for the codimension.
i ω |W.ω| mω
all p, n > 21 p ∤ n 6 20 p | n 6 20
2 ω2 + ωn
1
2
(n+ 1)n(n− 1) 1 1 1
1 ω1 n+ 1 > 1 n− 1 n− 2
Weight No. of l
strings strings all p, n > 21 p ∤ n 6 20 p | n 6 20
µ2
1
2
(n− 1)(n2 − 5n+ 8)
µ2 µ2
3
2
(n− 1)(n− 2) 3
2
(n− 1)(n− 2) 3
2
(n− 1)(n− 2) 3
2
(n− 1)(n− 2)
µ2 µ1 µ2 n− 1 n− 1 2(n− 1) 2(n− 1)
µ1 µ1 1 1 n− 1 n− 2
Lower bound on |Λ| 1
2
(3n2 − 7n+ 6) 3
2
n(n− 1) 1
2
(3n2 − 3n− 2)
Figure 5.4: λ = ω2 + ωn for n > 5
Certainly for all p we have |Λ| > 1
2
(3n2 − 3n − 2) and (†) is satisfied for n > 8.
Furthermore, for n ∈ [6, 7] since |Φ(G)| < |Λ| in these cases (♦) is satisfied for all X ∈ I.
If n = 5 and p 6= 5 then |Λ| > 30 and X = ∅ is the only centraliser type requiring further
consideration. The same is true for n = p = 5 since |Λ| > 29 here.
59
We use the fact that there are certainly two orthogonal roots not in Φs; take roots
α1, α5 and compute the 66 weights for this module explicitly. We provide weight nets
below. Note that we have omitted a factor of 1
6
on each coefficient.
1× 2 and 2× 1 nets: · 10 9 8 ·, · 4 9 2 ·, · -2 3 8 ·, · 4 -3 2 ·, · -2 3 -4 ·, · -8 -3 2 ·, · -2 -9 -4 ·, · -8 -9 -10 ·
2× 2 nets: · 4 9 8 ·, · 4 3 8 ·, · 4 3 -4 ·, · 4 -3 -4 ·, · -8 -3 -4 ·, · -8 -9 -4 ·
These weight nets do not involve any weights from W.ω1. We give below the weight
nets that do involve such weights (which we embolden):
5 4 3 2 -5 5 4 3 2 1 5 4 3 2 7
-1 4 3 2 -5 -1 4 3 2 1 -1 4 3 2 7
5 -2 -3 -4 -5 5 -2 -3 -4 1
-1 -2 -3 -4 -5 -1 -2 -3 -4 1
-7 -2 -3 -4 -5 -7 -2 -3 -4 1
5 -2 -3 2 1
-1 -2 -3 2 -5 -1 -2 -3 2 1 -1 -2 -3 2 7
-7 -2 -3 2 1
5 -2 3 2 1
-1 -2 3 2 -5 -1 -2 3 2 1 -1 -2 3 2 7
-7 -2 3 2 1
So we see that when p 6= 2, 5 we have |Λ| > 6 + 6 + 4 + 4 + 8 + 2.6 = 40, when p = 2
we have |Λ| > 42 (the increase derives from the 3 × 2 and 2 × 3 weight nets) and when
p = 5 we have |Λ| > 5 + 5 + 3 + 3 + 8 + 2.6 = 36; in any case |Λ| exceeds dimG, so we
are done.
Since ω2 + ωn < 2ω1 + ωn and the value of mω1 for L(2ω1 + ωn) is at least the
corresponding value of mω1 for L(ω2+ωn) when n 6 20, we can use Premet’s theorem 2.6
and the calculation of the number of weight strings in Figure 5.4 to conclude for n > 8
that (†) holds for L(2ω1+ωn). In fact we see in Figure 5.5 that (†) also holds for n ∈ [5, 7].
5.3 Centraliser analysis for L(ω4) and L(ω5)
In this section we examine the irreducible modules with highest weight ω4 for n ∈ [7, 12]
and ω5 for n = 9. For convenience we shall introduce some notation for weights in the
Weyl group orbit of a fundamental weight.
Notation Consider a fundamental weight ωk with 1 6 k 6 n written as a sum of simple
roots. If we write a fundamental weight as a string of the coefficients in such a sum
60
i ω |W.ω| mω
all p, n+ 2 > 23 2 6= p ∤ n+ 2 6 22 2 6= p | n+ 2 6 22
3 2ω1 + ωn (n+ 1)n 1 1 1
2 ω2 + ωn
1
2
(n+ 1)n(n− 1) 1 1 1
1 ω1 n+ 1 > 1 n n− 1
Weight No. of l
strings strings all p, n+ 2 > 23 2 6= p ∤ n+ 2 6 22 2 6= p | n+ 2 6 22
µ3 (n− 1)(n− 2)
µ3 µ3 n− 1 n− 1 n− 1 n− 1
µ3 µ2 µ3 n− 1 n− 1 n− 1 n− 1
µ3 µ1 µ1 µ3 1 2 n+ 1 n
µ2
1
2
(n− 1)(n− 2)(n− 3)
µ2 µ2
3
2
(n− 1)(n− 2) 3
2
(n− 1)(n− 2) 3
2
(n− 1)(n− 2) 3
2
(n− 1)(n− 2)
µ2 µ1 µ2 n− 1 n− 1 2(n− 1) 2(n− 1)
Lower bound on |Λ| 1
2
(3n2 − 3n+ 4) n
2
(3n+ 1) 1
2
(3n2 + n− 2)
Figure 5.5: λ = 2ω1 + ωn for n > 5
and imagine a zero coefficient on either end of this expression we see that we can encode
a fundamental weight by an ordered string of plus signs and minus signs depending on
whether the difference between successive coefficients is positive or negative. Thus ωk can
be written as k plus signs and n − k + 1 minus signs. The effect of the Weyl group is
to permute the signs in the string, so all weights in the Weyl group orbit of ωk can be
written as all possible strings of plus and minus signs as described.
We can see when two different weights in W.ωk differ by a root precisely if they have
k− 1 plus signs in the same position. Suppose we have two weights which (when written
in plus-minus notation) are identical except the first has a plus in the ith position and
the second in the jth position, then, assuming i < j, the difference between the first and
second weights is αi + · · · + αj−1. So, if two weights in different clusters have k − 1 plus
signs in the same position, then at most one of the clusters of weight spaces can lie in
Π(V ) \ Λ, provided the difference between the weightsis a linear combination of roots
outside Φs.
Later we shall compare sizes of clusters of weights in cliques; in this case it is helpful
to write a cluster as a string of plus and minus signs partitioned by bars and concluded
with a colon to indicate the centraliser type. Thus the bar is used to separate non-trivial
simple systems in X and the colon signifies that the signs which follow lie in positions
unaffected by reflection in any root in X. For example, if the centraliser type is A2A1
61
then ++−|+− : −−− denotes a cluster consisting of six weights lying in W.ω3 with n = 7,
i.e., the cluster consists of the weights ++−+−−−−, +−++−−−−, −+++−−−−, ++−−+−−−,
+−+−+−−− and −++−+−−−. As a shorthand we shall underline a string of plus and minus
signs in a cluster to denote all clusters which occur containing all possible arrangements
of the underlined symbols. For instance, if X = A2 we shall use ++− : +−−−− to denote
the five clusters with exactly one plus sign in the last five positions.
In the next lemma, we shall employ the following strategy. Suppose that G is of type
An acting on the G-module L(λ). The initial lower bound for |Λ| may be such that (♦)
is satisfied for all but a few subsystems X ∈ I of low rank but does not exceed dimG (so
we cannot immediately conclude that (†) holds). We can certainly assume that there are
two orthogonal roots outside of Φs for the remaining subsystems. We make the general
observation that either X = An−1 or there are two orthogonal roots outside Φs. If we
can show that d
An−1
λ > dimG and subsequently show that |Λ| > dimG by taking two
orthogonal roots outside Φs, then we can conclude that (†) holds.
Lemma 5.9. Let G act on the irreducible module V = L(ωk) for k ∈ [4, 5]. Then for
k = 4 when n ∈ [11, 12] and for k = 5 when n = 9 the condition (†) holds, otherwise (♦)
is satisfied for all X ∈ I.
Proof. Consider the module with highest weight ωk in An. We may assume that there
is a root α /∈ Φs so there are rAk−1An−k =
(
n−1
k−1
)
weights µ ∈ W.ωk with weight string of
the form µ, µ − α. By the adjacency principle we see that each weight string of length
2 contains at least one weight space that cannot lie in the eigenspace Vγ(s) for each
k ∈ [4, 5].
Consider the Weyl group orbit of ω4; it has size
(
n+1
4
)
. For n = 12 we have |Λ| >(
11
3
)
= 165. Since |Φ(G)| = 156 we see that (♦) is satisfied for all centraliser types and
the result holds. We wish to show that (†) holds, so we need to show that it holds both
when X = A11 and when there are two orthogonal roots outside Φs. (It is clear that the
only centraliser type where there are not two orthogonal roots outside Φs is X = A11.)
Suppose that X = A11 with simple roots α1, . . . , α11; there are two clusters, the first
62
containing weights whose last sign is plus and the second containing those whose last sign
is minus; so the clusters sizes are
(
12
3
)
= 220 and
(
12
4
)
= 495, whence dA11ω4 > 220 > dimG.
We can therefore assume that there are at least two orthogonal roots lying outside Φs;
take them to be α1 and α12. In particular, for n = 12, we have
ω4 = 9 18 27 36 32 28 24 20 16 12 8 4,
omitting a factor of 1
13
on each coefficient. As described above we can write this as a
string of four plus signs followed by nine minus signs and we obtain all possible strings
as the Weyl group interchanges plus signs and minus signs. Thus we find that we have
a convenient combinatorial approach to counting the number of weights orthogonal to α1
and α12, i.e., those weights which begin and end with either two plus signs or two minus
signs. Firstly we see that there are
(
11
2
)
+
(
11
4
)
weights orthogonal to α1: those having two
plus signs and those having two minus signs at the beginning of the string. Hence all other
weights are in pairs differing by α1, giving codimension |Λ| >
1
2
(715 − 385) = 165. (Of
course this agrees with our calculation above.) Then we see that there are 1 + 2
(
9
2
)
+
(
9
4
)
weights orthogonal to α1 and α12. All other weights occur in pairs differing by α1 or α12,
so, by the adjacency principle, |Λ| > 1
2
(715− 199) = 258 > dimG.
The case n = 11 follows similarly. We have |Λ| >
(
10
3
)
= 120, hence we are done if
|Φ(X)| > 12 by (♦). Suppose that X = A10 with simple roots α1, . . . , α10; there are two
clusters, the first containing weights whose last sign is plus and the second containing
those whose last sign is minus; so the clusters sizes are
(
11
3
)
= 165 and
(
11
4
)
= 330, whence
dA10ω4 > 165 > dimG.
Thus assume that there are two orthogonal roots lying outside Φs; take them to be
α1 and α11. We can now write ω4 as a string of four plus signs followed by eight minus
signs. We calculate that there are
(
10
2
)
+
(
10
4
)
weights in W.ω4 orthogonal to α1 giving
codimension |Λ| > 1
2
(495− 255) = 120 and there are 1 + 2
(
8
2
)
+
(
8
4
)
weights orthogonal to
α1 and α11. All other weights occur in pairs differing by α1 or α11, so, by the adjacency
principle, |Λ| > 1
2
(495− 127) = 184 > dimG.
63
For n = 10 we have |Λ| >
(
9
3
)
= 84 and we have the result when |Φ(X)| > 26 using
(♦). Thus there will certainly be two orthogonal roots not in Φs, namely α1 and α10.
There are 1+ 2
(
7
2
)
+
(
7
4
)
weights in W.ω4 orthogonal to both α1 and α10, all other weights
occurring in pairs differing by α1 or α10, hence |Λ| >
1
2
(330 − 78) = 126 > dimG. We
conclude that (♦) is satisfied for all X ∈ I.
For n = 9 we have |Φ(G)| = 90. From our initial calculation above we have |Λ| > 56
and (♦) shows that we may take two orthogonal roots α1, α9 not in Φs. This improves
our lower bound of the codimension of the eigenspace since there are 1 + 2
(
6
2
)
+
(
6
4
)
= 46
weights in W.ω4 orthogonal to both α1 and α9; we find that |Λ| >
1
2
((
10
4
)
− 46
)
= 82. As
a consequence, we may assume that there are three orthogonal roots outside Φs; taking
the third to be α5 we obtain |Λ| >
1
2
((
10
4
)
− (1 + 3
(
4
2
)
+ 3)
)
= 94, whence (♦) is satisfied
for all X ∈ I.
Now consider the module with highest weight ω5 when n = 9. We show that (†) holds.
We have |Λ| > 70 and so we are done if |Φ(X)| > 20 using (♦). Suppose that X = A8
with simple roots α1, . . . , α8; there are two clusters, the first containing weights whose
last sign is plus and the second containing those whose last sign is minus; so both clusters
have size
(
9
4
)
= 126, whence dA8ω5 > 126 > dimG.
Thus assume there are two orthogonal roots lying outside of the centraliser, take α1
and α9. We view weights in W.ω5 as ordered strings of five plus signs and five minus
signs. We count
(
6
1
)
+2
(
6
3
)
+
(
6
5
)
weights orthogonal to both α1 and α9, so |Λ| > 100. The
conclusion follows since dimG = 99.
It remains for us to consider L(ω4) when n ∈ [7, 8]. Note that dimL(ω4) = |W.ω4| = 70
or 126 according to whether n = 7 or 8.
Lemma 5.10. Let G act on the irreducible module V = L(ω4) for n = 8. Then (♦) holds.
Proof. Consider L(ω4) with n = 8. We know that |Λ| > rA3A4 =
(
7
3
)
= 35 and we are
done for all types of centralisers satisfying |Φ(X)| > 37. For the remaining types, there
will always be two orthogonal roots not lying in Φs. Assume that α1 and α3 are two
orthogonal roots outside Φs. Since there are 26 weights orthogonal to them both, we have
64
|Λ| > 1
2
(126 − 26) = 50. Repeating this procedure we are done for |Φ(X)| > 22 by (♦),
and for the remaining possibilities for X there are three orthogonal roots not in Φs, which
we may take to be α1, α3 and α5. There are 12 weights orthogonal to these roots, so
|Λ| > 57. Repeating this, we may take four orthogonal roots not in Φs and can improve
the lower bound to |Λ|. Indeed, we find that |Λ| > 60 and consequently we are done
if |Φ(X)| > 12. This leaves eleven centraliser types each requiring individual attention.
They are X = ∅, A1, A21, A
3
1, A
4
1, A2, A2A1, A2A
2
1, A2A
3
1, A
2
2 and A3.
For each type we arrange the weights into clusters and then form cliques. As described
in Section 4.2, in each clique the weight spaces corresponding to the weights constituting
at most one cluster can lie in the eigenspace. For each clique, the lower bound l for the
contribution to |Λ| is therefore the sum of all the cluster sizes except the largest. The
tables below show clusters arranged in cliques and calculate the minimal contribution l
of each clique to |Λ|. Note that we do not provide all clusters in the tables that follow,
only a selection. For each centraliser type X we find from the table that the codimension
exceeds eXω4 .
X = A3
Clique Cluster size l Clique Cluster size l
+++− : +−−−− 4 16 + +−− : −−++− 6 12
+ +−− : ++−−− 6 18 +−−− : + + +−− 4 8
+ +−− : −++−− 6 12
For X = A3, we see that d
A3
ω4
> 66 > 60 = eA3ω4 .
X = A2
2
Clique Cluster size l Clique Cluster size l
++−|+−− : +−− 9 18 + +−|++− : −−− 9 27
+−−|+−− : + +− 9 18 +−−|++− : +−− 9
For X = A22, we see that d
A2
2
ω4 > 63 > 60 = e
A2
2
ω4 .
X = A2A31
Clique Cluster size l Clique Cluster size l
++−|+−|+−| − − : 12 24 +−−|+−|++| − − : 6 6
+ +−|+−| − −|+− : 12 +−−| − −|++|+− : 6
+ +−| − −|+−|+− : 12 +−−|+−| − −|++ : 6 6
+−−|++|+−| − − : 6 6 +−−| − −|+−|++ : 6
+−−|++| − −|+− : 6
65
IfX = A2A
3
1 there is a cluster of weights +−−|+−|+−|+− : of size 24. If the weights in this
cluster lie in Π(V ) \Λ, then none of the 72 weights in the table above can lie in Π(V ) \Λ.
On the other hand, if this cluster lies in Λ then d
A2A31
ω4 > 24 + 42 = 66 > 60 = e
A2A31
ω4 .
X = A2A21
Clique Cluster size l Clique Cluster size l
++−|+−|+− : −− 12 24 + +−|++| − − : −− 3 6
+−−|+−|+− : +− 12 +−−|++| − − : +− 3
+ +−|+−| − − : +− 6 6 + +−| − −|++ : −− 3 6
+ +−| − −|+− : +− 6 6 +−−| − −|++ : +− 3
+−−|++ |+− : −− 6 6 −−−|+−|++ : +− 2 2
+−−|+− | − − : + + 6 6 −−−|++|+− : +− 2 2
For X = A2A
2
1, we see that d
A2A21
ω4 > 64 > 62 = e
A2A21
ω4 .
X = A2A1
Clique Cluster size l Clique Cluster size l
++−|+− : +−−− 6 18 + +−|++ : −−−− 3 12
+−−| − − : + + +− 3 9 +−−|++ : +−−− 3
+−−|+− : −++− 6 12 +−−|+− : ++−− 6 12
+ +−| − − : ++−− 3 6
For X = A2A1, we see that d
A2A1
ω4
> 69 > 64 = eA2A1ω4 .
X = A2
Clique Cluster size l
++− : + +−−−− 3 36
+−− : + + +−−− 3 48
For X = A2, we see that d
A2
ω4
> 84 > 66 = eA2ω4 .
X = A4
1
Clique Cluster size l Clique Cluster size l
+− |+−|+−|+− : − 16 32 +− |+−|++|−− : − 4 8
+− |+−|+−| − − : + 8 +− |+−| − −|++ : − 4 8
+ + |+− |+−| − − : − 4 8 + + |+− | − −| − − : + 2 4
+−|++|+− | − − : − 4 8
For X = A41, we see that d
A4
1
ω4 > 68 > 64 = e
A4
1
ω4 .
66
X = A3
1
Clique Cluster size l Clique Cluster size l
+− |+−|+− : +−− 8 16 +− | − −| − − : + + + 2 4
+ + |+−|+− : −−− 4 8 + + |+−| − − : +−− 2 4
+− |+−| − − : + +− 4 8 + + | − −|+− : +−− 2 4
+− | − −|+− : + +− 4 8 +− |++| − − : +−− 2 4
−− |+−|+− : + +− 4 8 −− |++|+− : +−− 2 4
For X = A31, we see that d
A3
1
ω4 > 68 > 66 = e
A3
1
ω4 .
X = A2
1
Clique Cluster size l Clique Cluster size l
+− |+− : + +−−− 4 32 −− |+− : + + +−− 2 16
+− | − − : + + +−− 2 16 + + |+− : +−−−− 2 8
For X = A21, we see that d
A2
1
ω4 > 72 > 68 = e
A2
1
ω4 .
X = A1 (and X = ∅)
Clique Cluster size l Clique Cluster size l
+− : ++ +−−−− 2 24 ++ : + +−−−−− 1 8
+− : −+++−−− 2 16 −− : + + ++−−− 1 8
+− : −−+++−− 2 8
It is clear that the lower bounds for dXω4 calculated in the table above are the same
whether X = ∅ or A1; in both cases we find dXω4 > 74 which exceeds both e
∅
ω4
= 72 and
eA1ω4 = 70.
Finally in this section we consider the module L(ω4) with n = 7.
Lemma 5.11. Let G act on the irreducible module V = L(ω4) for n = 7. Then (♦) is
satisfied for each X ∈ I \ {A23}.
Proof. We have dimL(ω4) = 70 for n = 7 and we can use the exact same iterative process
as we did for n = 8 to show that we are done provided that |Φ(X)| > 24. There are
seventeen centraliser types remaining to be investigated. They are (with eXω4 given in
brackets) as follows: ∅, A1, A21, A
3
1, A
4
1, A2, A2A1, A2A
2
1, A
2
2, A
2
2A1, A3, A3A1, A3A
2
1,
A3A2, A
2
3, A4 and A4A1.
67
X = A4A1
Clique Cluster size l Clique Cluster size l
++++−| − − : − 5 15 + +−−−|++ : − 10 10
+ ++−−|+− : − 20 + +−−−|+− : + 20
+ ++−−| − − : + 10 +−−−−|++ : + 5
In the table above we arrange the clusters into cliques and we see that dA4A1ω4 > 25. If
the 20 weights in the cluster ++−−−|+− : + are in Π(V )\Λ then dA4A1ω4 > 45 > 34 = e
A4A1
ω4
.
For, we exclude the weights in the clusters +++−−|+− : −, +++−−| − − : +, ++−−−|++ : −
and +−−−−|++ : + from the eigenspace since these clusters all contain at least one weight
differing in precisely two positions from at least one of the weights in + + − − −| + − : +. If
we do not have the weights in ++−−−|+− : + then the table above shows that dA4A1ω4 > 35.
If X = A4 the clusters that occur are easy to see from those given above. There
are two cliques (corresponding to the left and right hand sides of the table) which both
guarantee a contribution of at least 25 to |Λ|, hence dA4ω4 > 50 > 36 = e
A4
ω4
.
X = A3A2
Clique Cluster size l Clique Cluster size l
++−− |++− : − 18 18 + +++ | − −− : − 1 5
+ +−− |+−− : + 18 + ++− |+−− : − 12
+−−− |+++ : − 4 5 + ++− | − −− : + 4
+−−− |++− : + 12
−−−− |+++ : + 1
From the arrangement of clusters into cliques in the table above, we have dA3A2ω4 > 28.
If the weights in the cluster ++−−|++− : − are in Π(V )\Λ then the clusters +++−|+−− : −,
+−−−|+++ : − and +−−−|++− : + do not contribute to the eigenspace, hence dA3A2ω4 > 47 >
38 = eA3A2ω4 . We can conclude similarly for the weights in ++−− |+−− : +. If neither cluster
of size 18 contributes to the eigenspace, then dA3A2ω4 > 46.
X = A3A21
Clique Cluster size l Clique Cluster size l
++++ | − −| − − : 1 9 + +−− |+−|+− : 24 14
+ + +− |+−| − − : 8 + +−− | − −|++ : 6
+ ++− | − −|+− : 8 +−−− |+−|++ : 8
+ +−− |++| − − : 6 6 −−−− |++|++ : 1
+−−− |++|+− : 8
68
If the weights in the cluster of size 24 lie in Π(V )\Λ then, by excluding other clusters,
we find d
A3A21
ω4 > 44 > 40 = e
A3A21
ω4 . Otherwise, as can be seen from the table above,
d
A3A21
ω4 > 45.
X = A3A1
Clique Cluster size l Clique Cluster size l
++++ | − − : −− 1 9 + +−− |+− : +− 12 26
+ ++− |+− : −− 8 + +−− |+− : −+ 12
+ ++− | − − : +− 4 + +−− | − − : + + 6
+ ++− | − − : −+ 4 +−−− |+− : + + 8
+ +−− |++ : −− 6 8 −−−− |++ : + + 1
+−−− |++ : +− 4
+−−− |++ : −+ 4
From the table we see that dA3A1ω4 > 43 > 42 = e
A3A1
ω4
.
X = A3
Clique Cluster size l Clique Cluster size l
++++ : −−−− 1 13 +−−− : + + +− 4 13
+ ++− : +−−− 4 −−−− : + + ++ 1
+ +−− : ++−− 6 12
+ +−− : −++− 6 12
From the table we see that dA3ω4 > 50 > 44 = e
A3
ω4
.
X = A2
2
A1
Clique Cluster size l Clique Cluster size l
+++|+−− | − − : 3 3 +−−|+++ | − − : 3 2
+ + +| − − − |+− : 2 −−−|+++ |+− : 2
+ +−|++− | − − : 9 27 + +−| − −− |++ : 3 3
+ +−|+−− |+− : 18 +−−|+−− |++ : 9
+−−|++− |+− : 18 −−−|++− |++ : 3
If we assume in turn that the weights in the clusters of size 18 are in Π(V ) \ Λ, we
obtain d
A2
2
A1
ω4 > 46 > 42 = e
A2
2
A1
ω4 for each. If these two clusters do not contribute to the
eigenspace then we have d
A2
2
A1
ω4 > 43 as required.
X = A2
2
Clique Cluster size l Clique Cluster size l
+++|+−− : −− 3 21 +−−|++− : +− 9 21
+ +−|++− : −− 9 +−−|++− : −+ 9
++−|+−− : +− 9 +−−|+−− : + + 9
+ +−|+−− : −+ 9 −−−|++− : + + 3
+−−|+++ : −− 3 2 + ++| − −− : +− 1 1
−−−|+++ : +− 1
69
From the table we see that d
A2
2
ω4 > 45 > 44 = e
A2
2
ω4 .
X = A2A21
Clique Cluster size l Clique Cluster size l
+++|+−| − − : − 2 3 + +−|+−| − − : + 6 6
+ ++| − −|+− : − 2 + +−| − −|+− : + 6
+ ++| − −| − − : + 1 + +−| − −|++ : − 3 3
+ +−|++| − − : − 3 3 +−−| − −|++ : + 3
+−−|++| − − : + 3 + +−|+−|+− : − 12 24
−−−|++|++ : − 1 3 +−−|++|+− : − 6
−−−|++|+− : + 2 +−−|+−|++ : − 6
−−−|+−|++ : + 2 +−−|+−|+− : + 12
If either of the clusters of size 12 are in Π(V )\Λ, we see that d
A2A21
ω4 > 49 > 46 = e
A2A21
ω4 .
Therefore, assuming that these weights are in Λ we have that the minimum contribution
of the largest clique above becomes 30. Thus d
A2A21
ω4 > 48.
X = A2A1
Clique Cluster size l Clique Cluster size l
+++|+− : −−− 2 3 +−−|++ : +−− 3 6
+ ++| − − : +−− 1 +−−|+− : + +− 6 15
+ +−|++ : −−− 3 15 +−−| − − : + + + 3
+ +−|+− : +−− 6 −−−|++ : + +− 1 3
+ +−| − − : + +− 3 6 −−−|+− : + + + 2
From the table above we have dA2A1ω4 > 48 = e
A2A1
ω4
. If the weights in the cluster
+++|+− : −−− are in Π(V )\Λ, then none of the clusters ++−|+− : +−− can lie in Π(V )\Λ,
so dA2A1ω4 > 51. Otherwise d
A2A1
ω4
> 49 and we are done.
X = A2
Clique Cluster size l Clique Cluster size l
+++ : +−−−− 1 4 +−− : + + +−− 3 6
+ +− : ++−−− 3 9 +−− : +−++− 3 6
+ +− : −++−− 3 6 +−− : −+++− 3 9
+ +− : −−++− 3 6 −−− : + + ++− 1 4
From the table, by arranging the clusters into cliques, we have dA2ω4 > 50 = e
A2
ω4
. We
can improve the codimension by 3 by observing that at most two of the 10 clusters from
+ − − : + + +−− correspond to weights lying in Π(V ) \ Λ. (In the table we were assuming
that at most three such clusters were contributing to the eigenspace dimension.) If we
assume without loss that the weights in the cluster + − − : + + + − − lie in Π(V ) \ Λ, this
70
forces all clusters but those of the form +−− : +−−++ to lie outside the eigenspace. But
only one of the clusters in +−− : +−−++ can contribute to the eigenspace.
X = A4
1
Clique Cluster size l Clique Cluster size l
++ |+−|+−| − − : 4 12 −− |++|+−|+− : 4 8
+− |++|+−| − − : 4 −− |+−|++|+− : 4
+− |+−|++| − − : 4 −− |+−|+−|++ : 4
+− |+−|+−|+− : 16 + + |++| − −| − − : 1
+ + |+−| − −|+− : 4 8 + + | − −|++| − − : 1
+− |++| − −|+− : 4 + + | − −| − −|++ : 1
+− |+−| − −|++ : 4 −− |++|++| − − : 1
+ + | − −|+−|+− : 4 8 −− |++| − −|++ : 1
+− | − −|++|+− : 4 −− | − −|++|++ : 1
+− | − −|+−|++ : 4
If the cluster of size 16 lies in Π(V ) \ Λ, then all clusters with size greater than one
cannot be in Π(V ) \ Λ, so d
A4
1
ω4 > 48 = e
A4
1
ω4 . We are done, unless the six weight spaces
corresponding to the remaining six clusters are in Π(V ) \ Λ. However, this implies that
2α4(s) = 2α6(s) = 1, i.e., α4(s) = α6(s) = −1 for s ∈ SA4
1
, so α4 + α5 + α6 ∈ Φs and
X 6= A41. Now using the table above and assuming that the cluster of size 16 does not
contribute to the dimension of the eigenspace, we have d
A4
1
ω4 > 48 and we improve the value
of the codimension by at least one using the previous observation about the six weights
constituting the clusters of size one.
X = A3
1
Clique Cluster size l Clique Cluster size l
++ |+−|+− : −− 4 20 +− |+−| − − : + + 4 8
+− |++|+− : −− 4 +− | − −|+− : + + 4
+− |+−|++ : −− 4 −− |+−|+− : + + 4
+− |+−|+− : +− 8 −− |++|+− : +− 2 2
+− |+−|+− : −+ 8 −− |+−|++ : +− 2
+ + |+−| − − : +− 2 2 −− |++|+− : −+ 2 2
+− |++| − − : +− 2 −− |+−|++ : −+ 2
++ |+−| − − : −+ 2 2 + + |++| − − : −− 1
+− |++| − − : −+ 2 ++ | − −|++ : −− 1
+ + | − −|+− : +− 2 2 + + | − −| − − : + + 1
+− | − −|++ : +− 2 −− |++|++ : −− 1
+ + | − −|+− : −+ 2 2 −− |++| − − : + + 1
+− | − −|++ : −+ 2 −− | − −|++ : + + 1
71
From the table above, we see that d
A3
1
ω4 > 40. By similar arguments as for the case
X = A41 we are done if any one of the clusters of sizes 8 or 4 contribute to the dimension
of the eigenspace. Hence, assuming that none of these contribute, it is easy to show
d
A3
1
ω4 > 51 > 50 = e
A3
1
ω4 .
X = A2
1
Clique Cluster size l Clique Cluster size l
++ |++ : −−−− 1 7 −− |++ : + +−− 1 4
+ + |+− : +−−− 2 +− | − − : + + +− 2 6
+− |++ : +−−− 2 6 −− |+− : + + +− 2 7
+− |+− : + +−− 4 16 −− | − − : + + ++ 1
+ + | − − : + +−− 1 4
From the table above we see that d
A2
1
ω4 > 50. Suppose that the two clusters in the clique
+ − | + − : + +−− lie in Π(V ) \ Λ; otherwise we are done since e
A2
1
ω4 = 52. Then the cliques
++ |+− : +−−− and +− |++ : +−−− do not contribute to the dimension of the eigenspace,
so d
A2
1
ω4 > 53.
If X = A1, we may separate the weights occurring into three types: there are
(
6
2
)
= 15
weights of the form ++ : + +−−−− ,
(
6
3
)
= 20 pairs of weights in clusters of the form
+− : + + +−−− and
(
6
2
)
= 15 weights of the form −− : + + ++−− . We wish to find an upper
bound for the dimension of the eigenspace. The first and third types can have at most
three weights lying in Π(V ) \ Λ. The second type can contribute at most eight weights
(from four clusters) to the dimension of the eigenspace. In particular, assume by using
the Weyl group that the cluster +− : +++−−− contributes to the eigenspace, in which case
it rules out any cluster with two plus signs in the third, fourth and fifth positions. (If
we have +− : −−−+++ also, then this rules out all of the remaining weights of the second
type, so only two clusters contribute in this case.) From the remaining weights suppose
without loss of generality that the cluster +− : +−−++− also contributes to the eigenspace.
This rules out all but four clusters and we can have at most the weights of two of these
lying in Π(V ) \Λ, i.e., we either have the two clusters +− : −+−+−+ and +− : −−+−++ or
+− : −−++−+ and +− : −+−−++ contributing to the eigenspace. Hence there are at most
eight weights in Π(V ) \ Λ and dA1ω4 > 62 > 54 = e
A1
ω4
.
72
Finally, suppose that X = ∅. In order to conclude that (♦) holds we shall show that
|Π(ω4) \ Λ| < 14 since e
∅
ω4
= 56. Indeed, we see from the 14 cliques below of size 5 that
|Π(V ) \ Λ| 6 14.
+ + + + −−−− + + −− + + −− + − + − + − + − + + + −− + −− + − + + − + −− + −− + + + −− + + − + −− + −
+ + + − + −−− + + −− + − + − + − + − + −− + + + + −−−− + + − + + −− + − + −− + + −− + + + − + −−− +
+ + − + + −−− + + −−− + + − + − + −−− + + + + −−− + − + + − + −− + + − + −− + − + − + + + −−−− + +
+ − + + + −−− + −−− + + + − + −−− + − + + + − + −− + − + + −− + − + + − + −−− + + − + + −− + −− + +
− + + + + −−− − + −− + + + − −− + − + − + + − + + −− + − + −− + + − + + − −−− + + + − + − + − + −− + +
−−−− + + + + −− + + −− + + − + − + − + − + −−− + + − + + − + −− + − + + − + + −−− + + −− + − + + − +
−−− + − + + + −− + + − + − + − + − + − + + − −−− + + + + − − + −− + + − + − + + −− + + − −− + − + + + −
−− + −− + + + −− + + + −− + − + − + + + −− −− + + + − + − − + − + + −− + − + + − + − + − −− + + + + −−
− + −−− + + + − + + + −−− + − + + + − + −− − + − + + − + − − + + − + −− + − + + + −− + − − + + − + + −−
+ −−−− + + + + − + + −−− + + + − + − + −− + −− + + − + − + + −− + −− + + + + −−− + − + − + − + + −−
Suppose without loss of generality that + + + + − − − − is in Π(V ) \ Λ. This rules out
16 weights which have three plus signs in any of the first four positions, i.e., the weights
+++− +−−− . We shall consider separately the two cases that the weight − − − − + + + +
either lies in or not in the eigenspace.
Suppose first that − − − − + + + + lies in Π(V ) \ Λ. As before, this rules out 16 more
weights, namely +−−− +++− . There are 36 weights remaining, all being of the form
++−− ++−− . Suppose without loss that ++−−++−− lies in Π(V )\Λ. This rules out the
8 weights + + − − +− +− and +− +− + + − −. Looking at the cliques above, we must have
the weight −−++++−− in Π(V ) \Λ which rules out −−++−−++ whence ++−−−−++ is
ruled out also. There are now 6 cliques (the 1st, 2nd, 7th, 8th, 9th and 14th reading left-
to-right above) with one weight present in Π(V ) \Λ and there are two weights remaining
in each of the other 8 cliques. The 6 weights that we are assuming are present in Π(V )\Λ
are invariant under H = 〈wα1 , wα3 , wα5 , wα7〉 which has a single orbit on the remaining
weights. Therefore by applying a suitable element of H we may assume that +−+−+−+−
lies in Π(V ) \ Λ. This determines which weights in the other seven pairs may lie in
Π(V ) \Λ. Thus we may have the following configuration of 14 weights lying in Π(V ) \Λ.
+ + + + −−−− + + −− + + −− + + −−−− + + + − + − + − + − + − + −− + − + + −− + − + + − + −− + + −− +
−−−− + + + + −− + + −− + + −− + + + + −− − + − + − + − + − + − + + − + − − + + − + −− + − + + −− + + −
Suppose that s = diag(λ1, . . . , λ8) ∈ S∅∩T . Since the pairs ++−−++−− and ++−−−−++,
+−+−+−+− and +−+−−+−+, and −++−+−−+ and −++−−++− are in Π(V )\Λ we have
(α5 + 2α6 + α7)(s) = (α5 + α7)(s) = (α5 − α7)(s) = 1.
73
Thus 2α6(s) = 2α7(s) = 1 implying that α6(s) = α7(s) = −1 since by assumption
α6(s) 6= 1. It follows that the root α6 + α7 ∈ Φs, hence the configuration of weights is
not possible. It follows that if a given weight is present in Π(V ) \ Λ, then its antipodal
weight where plus and minus signs are interchanged cannot lie in Π(V ) \ Λ.
Now we are assuming that −−−−++++ lies in Λ. If no weight in Π(V )\Λ has two plus
signs in two of the first four positions, i.e., all weights ++−− ++−− are in Λ then the 4th,
5th, 7th, 9th, 10th and 13th cliques above are ruled out, so there are at most 8 weights
in Π(V ) \ Λ. Thus we may assume that we must have at least one weight of this form,
take ++−−++−− . This rules out 13 weights −−++−−++ , ++−− +− +− , +− −−++ +−
and +− +− ++−−.
Suppose we have no weight of the form +− −− +− ++. Then there are only two weights
not ruled out in each of the 4th, 8th and 10th cliques above. Consider the 8th clique. We
are done if we have no weights in this clique in Π(V ) \ Λ so assume that we have one of
them. If −−−+−+++ is in Π(V ) \Λ then the two weights in the 10th clique cannot be in
Π(V ) \Λ. Similarly, if −−+−−+++ is in Π(V ) \Λ then the two weights in the 4th clique
cannot be in Π(V ) \ Λ. We may assume therefore, since H preserves the weights already
in Π(V ) \ Λ and has only one orbit on the remaining weights, that + − − − − + + + lies in
Π(V ) \ Λ. The following 26 weights are not ruled out from lying in Π(V ) \ Λ.
− + −− + − + + − + + − − + + − − + + − −− + + −− + + + − + −
−− + + + + −− −− + − + + + − −− + + − + + − −− + − + − + +
+ − + − + − + −
If no weights of the form +− +− + − +− lie in Π(V ) \ Λ then there are at most 12
weights in Π(V ) \ Λ: 3 from the three weights we are assuming are present, at most 2
from the 2nd column, at most 2 from both sets of weights in the 3rd column above, at
most 1 from both sets of weights in the 4th column and at most 1 from the 5th column.
Therefore we must have at least one of these 8 weights present in Π(V )\Λ. The subgroup
J = 〈wα3 , wα7〉 preserves the set of weights already in Π(V ) \ Λ and has an orbit of size
2 on the remaining weights. Thus we may assume by applying appropriate elements of J
that either + − + − + − + − or − + + − + − + − lies in Π(V ) \ Λ. In the former case, there are
20 weights remaining that have not been ruled out from being in Π(V ) \ Λ. In the 4th
74
clique above only the weight −++−−+−+ remains which we may assume lies in Π(V ) \Λ,
but this means that no weight in the 13th clique can lie in Π(V ) \ Λ. The latter case is
slightly more difficult. There are 17 weights remaining which can be rearranged in eight
cliques as follows.
+ − + − + −− + + −− + + − + − − + − + − + + − − + + −− + − + − + − + + −− + −− + + −− + + −− + + + + −−
+ −− + + −− + −−− + + + + − −− + + − + + − − + − + − + − + − + − + −− + + −−− + + − + + −− + + + −− +
−− + + − + − +
−− + − + + − +
−−− + + + − +
Hence, assuming that ++++−−−−, ++−−++−−, +−−−−+++ and −++−+−+− lie in
Π(V ) \Λ together with one weight from each of the above eight cliques, we are done.
We are unable to show that d
A2
3
ω4 > 33 here. The clusters are shown in the table below.
It may be the case that the clusters of size 36 and both of size 1 lie in Π(V ) \Λ, in which
case we can only conclude that d
A2
3
ω4 = 32 and 2α4(s) = 1 whence α4(s) = −1. The set
SA2
3
consists of matrices s = diag(a, a, a, a, b, b, b, b) for a, b ∈ K∗ with a4b4 = 1 and, as
just mentioned, it may be the case that a = −b.
X = A2
3
Cluster Cluster size
+ + ++ | − − −− : 1
+ + +− |+−−− : 16
+ +−− |++−− : 36
+−−− |+++− : 16
−−−− |++++ : 1
5.4 Centraliser analysis for L(ω3)
In this section we shall consider the modules L(ω3) for n > 8, L(ω1 + ω2) for n > 5 and
L(3ω1) for n > 5.
First, suppose that the largest rank of a component in the root subsystem X ∈ I is at
least n−k+1 where n > 2k−2. We claim that there are at most k−1 orthogonal roots not
in Φs. The possible centraliser types are determined by taking subsystems of the extended
Dynkin diagram. Suppose that the root subsystem of largest rank in X is An−k+1 where
X contains the simple roots αi+1, . . . , αn−j for 0 6 i 6 k − 1 and j = k − i − 1. Then
75
either αi /∈ Φs for i > 1 or αn−j+1 /∈ Φs for j > 1. A maximal set of orthogonal roots not
in Φs is
αi αn−j+1
αi−1 + αi + αi+1 αn−j + αn−j+1 + αn−j+2
...
...
α1 + · · ·+ αi + · · ·+ α2i−1 αn−2j+2 + · · ·+ αn−j+1 + · · ·+ αn
There are i+ j = k − 1 such roots.
Lemma 5.12. Let G act on the irreducible module L(ω3) for n > 14. Then (♦) holds for
each X ∈ I.
Proof. For a given n, each weight in the Weyl group orbit of ω3 can be written as an
ordered string of 3 plus signs and n − 2 minus signs; there are
(
n+1
3
)
such weights. The
number of weights in W.ω3 orthogonal to α1 is n−1+
(
n−1
3
)
so that |Λ| > 1
2
(n−1)(n−2).
There are two orthogonal roots outside Φs provided X 6= An−1. If X = An−1 with
simple roots α1, . . . , αn−1 then there are two clusters, the first containing weights whose
last sign is plus and the second containing those whose last sign is minus; so the cluster
sizes are
(
n
2
)
and
(
n
3
)
, whence dAn−1ω3 >
(
n
2
)
> 2n = eAn−1ω3 . Therefore, in this case, (♦)
holds for n > 6. There are 2(n− 3) +
(
n−3
3
)
weights orthogonal to both α1 and α3 so we
obtain an improved lower bound for |Λ|; we have |Λ| > n2 − 5n+ 8.
There are three orthogonal roots not in Φs provided X 6= An−2 or a root subsystem
properly containing it. However, eAn−2ω3 = 2(2n − 1) so (♦) holds in this case for n > 8.
Taking pairs of weights differing by α1, α3 or α5 we find that |Λ| >
1
2
(3n2 − 21n+ 50).
There are four orthogonal roots not in Φs provided X 6= An−3 or a root subsystem
properly containing it. However, eAn−3ω3 = 6(n− 1) so (♦) holds in this case for n > 9. We
now find that |Λ| > 2(n2 − 9n+ 28) by considering orthogonal roots α1, α3, α5 and α7.
We may take five orthogonal roots outside Φs provided X 6= An−4 or a root subsystem
76
properly containing it. Since eAn−4ω3 = 4(2n−3) we find that (♦) holds for n > 10. There are
5(n−9)+
(
n−9
3
)
weights orthogonal to α1, α3, α5, α7 and α9 so that |Λ| >
5
2
(n2−11n+42)
which exceeds dimG for n > 16.
We may take a sixth orthogonal root outside the centraliser since eAn−5ω3 = 10(n−2) so
that (♦) is satisfied for X = An−5 for n > 11 or a root subsystem properly containing it.
Thus assuming that the six orthogonal roots α1, α3, α5, α7, α9 and α11 are not in Φs we
find that |Λ| > 1
2
[(
15
3
)
− 6.3− 1
]
= 218 or 1
2
[(
16
3
)
− 6.4− 4
]
= 266 according as n = 14
or 15. Thus (♦) holds for all X ∈ I when n ∈ [14, 15].
We next consider L(ω3) for n ∈ [8, 13]. Here we shall be forced to consider in detail
the weights in W.ω3 for many different centraliser types.
Lemma 5.13. Let G act on the irreducible module L(ω3) for n ∈ [8, 13]. Then (♦) holds
for each X ∈ I when n ∈ [9, 13] and for each X ∈ I \ {A32, A4A3} when n = 8.
Proof. If n ∈ [12, 13] then, from the previous lemma, we can take six orthogonal roots
not in Φs. We see that |Λ| >
1
2
[(
13
3
)
− 6
]
= 140 for n = 12 and |Λ| > 1
2
[(
14
3
)
− 6.2
]
= 176
for n = 13. Thus (♦) holds provided |Φ(X)| > 16 or 6 according as n = 12 or 13.
If n = 10 or 11 we may take five orthogonal roots outside Φs in which case |Λ| >
1
2
[(
11
3
)
− 5
]
= 80 if n = 10 or 1
2
[(
12
3
)
− 10
]
= 105 if n = 11. Thus (♦) holds if |Φ(X)| >
110− 80 = 30 for n = 10 or if |Φ(X)| > 132− 105 = 27 for n = 11.
Similarly for n = 9 we may take four orthogonal roots α1, α3, α5 and α7 not in Φs.
The module has dimension
(
10
3
)
= 120 and there are 8 weights orthogonal to the four
roots. Thus |Λ| > 56 and (♦) holds provided |Φ(X)| > 90− 56 = 34.
If n = 8 we wish to take four orthogonal roots not in Φs. We require (♦) to hold
for X = A5 and root subsystems properly containing it. We can take three orthogonal
roots, say α1, α3 and α5 to show that |Λ| >
1
2
[(
9
3
)
− 10
]
= 37 so (♦) holds provided
|Φ(X)| > 72− 37 = 35. Therefore we need only consider X = A5 and A5A1, else |Φ(X)|
is large enough. If X = A5 then the two cliques + + + − −− : − − − , + + − − −− : +−− and
+−−−−− : + +− , −−−−−− : +++ show that dA5ω3 > 58 > 42 = e
A5
ω3
. If X = A5A1 then we may
again form two cliques indicated in the table below to show that dA5A1ω3 > 42 > 40 = e
A5A1
ω3
.
77
(λ, n, X) = (ω3, 8, A5A1)
Cliques Cluster size l Cliques Cluster size l
+++−−− | − − : − 20 35 +−−−−− |++ : − 6 7
+ +−−−− |+− : − 30 +−−−−− |+− : + 12
+ +−−−− | − − : + 15 −−−−−− |++ : + 1
Finally we may also take α7 /∈ Φs and we see that |Λ| >
1
2
(84 − 4) = 40 whence (♦)
holds if |Φ(X)| > 72− 40 = 32. There are 25 centraliser types to consider.
The following table lists the centraliser types requiring consideration for L(ω3) with
n ∈ [8, 13].
n Centraliser types
13 ∅, A1, A21, A
3
1, A2
12 ∅, A1, A21, A
3
1, A
4
1, A
5
1, A
6
1; A2, A2A1, A2A
2
1, A2A
3
1, A2A
4
1, A2A
5
1, A
2
2,
A22A1, A
2
2A
2
1; A3, A3A1, A3A
2
1
11 ∅, A1, A21, A
3
1, A
4
1, A
5
1, A
6
1; A2, A2A1, A2A
2
1, A2A
3
1, A2A
4
1, A
2
2, A
2
2A1,
A22A
2
1, A
2
2A
3
1, A
3
2, A
3
2A1, A
4
2; A3, A3A1, A3A
2
1, A3A
3
1, A3A
4
1, A3A2, A3A2A1,
A3A2A
2
1, A3A
2
2, A3A
2
2A1, A
2
3, A
2
3A1; A4, A4A1, A4A
2
1, A4A
3
1, A4A2
10 ∅, A1, A21, A
3
1, A
4
1, A
5
1; A2, A2A1, A2A
2
1, A2A
3
1, A2A
4
1, A
2
2, A
2
2A1,
A22A
2
1, A
3
2, A
3
2A1; A3, A3A1, A3A
2
1, A3A
3
1, A3A2, A3A2A1, A3A2A
2
1,
A3A
2
2, A
2
3, A
2
3A1, A
2
3A2; A4, A4A1, A4A
2
1, A4A
3
1, A4A2, A4A2A1; A5
9 ∅, A1, A21, A
3
1, A
4
1, A
5
1; A2, A2A1, A2A
2
1, A2A
3
1, A
2
2, A
2
2A1, A
2
2A
2
1, A
3
2;
A3, A3A1, A3A
2
1, A3A
3
1, A3A2, A3A2A1, A3A
2
2, A
2
3, A
2
3A1; A4, A4A1,
A4A
2
1, A4A2, A4A2A1, A4A3; A5, A5A1, A5A
2
1
8 ∅, A1, A21, A
3
1, A
4
1; A2, A2A1, A2A
2
1, A2A
3
1, A
2
2, A
2
2A1, A
3
2; A3, A3A1,
A3A
2
1, A3A2, A3A2A1, A
2
3; A4, A4A1,A4A
2
1, A4A2, A4A3; A5, A5A1
Table 5.2: Centraliser types requiring consideration for L(ω3) with n ∈ [8, 13]
We shall now consider the five centraliser types remaining for n = 13.
Suppose X = A2 with simple roots α1 and α2. There is a contribution of at least 30 to
|Λ| from the clique ++− : +−−−−−−−−−−. There can be at most five clusters with one
plus sign in the first three positions present in Π(V )\Λ, for example +−− : ++−−−−−−−−−,
+−− : −−++−−−−−−−, +−− : −−−−++−−−−−, +−− : −−−−−−++−−− and +−− : −−−−−−−−++−.
The remaining clusters of this form therefore contribute at least 150 to |Λ| so in total we
have dA2ω3 > 180 > 176 = e
A2
ω3
.
Suppose X = A31 with simple roots α1, α3 and α5. Consider cliques with one plus sign
in the first six positions. Assuming that it is in one of the first two positions there can be
78
at most four cliques of this form lying in Π(V ) \Λ, for example +− |−−|−− : ++−−−−−− ,
+ − | − −| − − : − − + + − − − − , + − | − −| − − : − − − − + + −− and + − | − −| − − : − − − − − − ++.
The same is true taking the plus sign in the 3rd and 5th position so there are at least[(
8
2
)
− 4
]
.2.3 = 144 weights in |Λ|. The cliques +− |+−| − − : +−−−−−−− ,
+−|−−|+− : +−−−−−−− and −−|+−|+− : +−−−−−−− contribute 7.4.3 = 84 to |Λ|, hence
(♦) holds since d
A3
1
ω3 > 228 > 176 = e
A3
1
ω3 .
Suppose X = A21 with simple roots α1 and α3; we have e
A2
1
ω3 = 178. Consider the
clusters with one plus sign in one of the first two positions and none in the 3rd and 4th
position. There can be at most five clusters lying in |Λ| since there are at most five distinct
weights of this form which do not have two plus signs in the same position. For example
we could take +− |−− : ++−−−−−−−− , +− |−− : −−++−−−−−− , +− |−− : −−−−++−−−− ,
+−|−− : −−−−−−++−− and +−|−− : −−−−−−−−++. Thus there are at least 80 weights in
|Λ| which we can double by considering clusters with one plus sign in one of the 3rd and
4th positions and none in the 1st and 2nd positions. The clique +− |+− : +−−−−−−−−−
contributes a further 36 weights to |Λ|. Thus we have d
A2
1
ω3 > 196 > 178 = e
A2
1
ω3 .
Suppose X = A1 with simple root α1. There can be at most six cliques with one
plus sign in the first two positions and two plus signs in the remaining positions lying in
Π(V ) \ Λ. There are 66 cliques in total of this form so |Λ| > 60.2 = 120. The collections
of cliques −− : +−− + +−−−−−−− , −− : +−−− + +−−−−−− and −− : +−−−− + +−−−−−
show that there are at least a further 3.7 + 4.6 + 5.5 = 70 weights in |Λ|. Thus we have
dA1ω3 > 190 > 180 = e
A1
ω3
.
Suppose X = ∅. The following table shows that d∅ω3 > 197 > 182 = e
∅
ω3
. We shall use
the notation +−− + +−−−−−−−−− to denote the three cliques +−− + +−−−−−−−−− ,
−+− + +−−−−−−−−− and −−+ + +−−−−−−−−− . The same notation is employed for
the other collections of cliques.
(λ, n, X) = (ω3, 13, ∅)
Collection of cliques Cluster size l Collection of cliques Cluster size l
+−− + +−−−−−−−−− 1 27 +−−−−− + +−−−−−− 1 36
+−−− + +−−−−−−−− 1 32 +−−−−−− + +−−−−− 1 35
+−−−− + +−−−−−−− 1 35 +−−−−−−− + +−−−− 1 32
79
The remaining cases for n ∈ [9, 12] are similar, and Appendix A contains the details
for n = 12. If n = 8 we can show that (♦) holds for all X ∈ I \ {A32, A4A3}; the argument
for X = ∅ is particularly involved. If X is either A32 or A4A3 then it is possible that
dXω3 = e
X
ω3
. In the former case the clusters are given in the table below. It is possible for
the cluster of size 27 and the three clusters of size 1 to lie inside the eigenspace so that
d
A3
2
ω3 = 54 = e
A3
2
ω3 .
(λ, n, X) = (ω3, 8, A32)
Clusters Cluster size Clusters Cluster size
+ + +| − − − | − −− : 1 +−−| − −− |++− : 9
+ +−|+−− | − −− : 9 −−−|+++ | − − − : 1
+ +−| − −− |+−− : 9 −−−|++− |+−− : 9
+−−|++− | − −− : 9 −−−|+−− |++− : 9
+−−|+−− |+−− : 27 −−−| − −− |+++ : 1
In the latter case there are just four clusters +++−−|−−−− :, ++−−−|+−−− :, +−−−−|++−− :
and −−−−−|+++− : of size 10, 40, 30 and 4, respectively. The second and fourth cluster
may lie inside the eigenspace, ruling out the other clusters by the adjacency principle, so
dA4A3ω3 = 40 = e
A4A3
ω3
.
We complete this section by considering both L(ω1 + ω2) and L(3ω1) for n ∈ [5,∞).
Lemma 5.14. Let G act on the irreducible modules L(ω1 + ω2) and L(3ω1) (p > 3) for
n ∈ [5,∞). Then (♦) holds for each X ∈ I.
Proof. The result follows for both modules for n ∈ [9,∞) by Premet’s theorem and the
previous lemmas since ω3 < ω1 + ω2 < 3ω1.
Throughout this lemma for convenience we shall omit a factor of 1
n+1
on the coefficients
of weights. From Lu¨beck’s tables [18] we find that mω3 = 1 or 2 according as p = 3 or
p 6= 3.
Consider L(ω1 + ω2) for n = 8. Recall the calculation above for L(ω3) which shows
that we can take four roots outside Φs. We may use this if p 6= 3 for L(ω1 + ω2) to
conclude that |Λ| > 80 so (♦) is satisfied. If p = 3 we see from Figure 5.6 that |Λ| > 36.
Therefore we may assume that there are two orthogonal roots not in Φs, say α1 and α8.
80
i ω |W.ω| mω
p 6= 3 p = 3
2 ω1 + ω2 n(n+ 1) 1 1
1 ω3
`
n+1
3
´
2 1
Weight No. of l
strings strings p 6= 3 p = 3
µ2 (n− 1)(n− 2)
µ2 µ2 n n n
µ2 µ1 µ2 n− 1 2(n− 1) n− 1
µ1 (n− 1)(n− 2)(n− 3)
µ1 µ1
1
2
(n− 1)(n− 2) (n− 1)(n− 2) 1
2
(n− 1)(n− 2)
Lower bound on |Λ| n2 1
2
n(n+ 1)
Figure 5.6: λ = ω1 + ω2 for n ∈ [3, 20]
The arrangements of weights below (with a factor of 1
9
omitted on each coefficient) and
the same arrangements with signs negated and coefficients reversed show that |Λ| > 44.
15 21 18 15 12 9 6 3 6 3 18 15 12 9 6 3 6 3 9 15 12 9 6 3 6 3 0 15 12 9 6 3 6 3 9 6 12 9 6 3 6 3 0 6 12 9 6 3
6 21 18 15 12 9 6 3 -3 3 18 15 12 9 6 3 -3 3 9 15 12 9 6 3 -3 3 0 15 12 9 6 3 -3 3 9 6 12 9 6 3 -3 3 0 6 12 9 6 3
-3 12 9 6 3 0 -3 3 6 12 9 6 3 0 -3 3 15 12 9 6 3 0 -3 3
-3 12 9 6 3 0 -3 -6 6 12 9 6 3 0 -3 -6 15 12 9 6 3 0 -3 -6
-3 3 9 6 3 0 -3 3 6 3 9 6 3 0 -3 3 6 3 9 6 3 9 6 3
-3 3 9 6 3 0 -3 -6 6 3 9 6 3 0 -3 -6 -3 3 9 6 3 9 6 3
6 3 0 -3 3 0 6 3 6 3 0 6 3 0 6 3 6 3 0 -3 -6 9 6 3 6 3 0 -3 3 9 6 3 6 3 0 -3 12 9 6 3 6 3 0 6 3 9 6 3
-3 3 0 -3 3 0 6 3 -3 3 0 6 3 0 6 3 -3 3 0 -3 -6 9 6 3 -3 3 0 -3 3 9 6 3 -3 3 0 -3 12 9 6 3 -3 3 0 6 3 9 6 3
6 3 9 6 3 0 6 3 -3 -6 9 6 3 0 -3 3 -3 3 0 6 3 0 -3 3 6 3 0 6 3 0 -3 3
-3 3 9 6 3 0 6 3 -3 -6 9 6 3 0 -3 -6 -3 3 0 6 3 0 -3 -6 6 3 0 6 3 0 -3 -6
Thus (♦) is satisfied for X = A5 and larger subsystems of A8 and we may assume
that there are four orthogonal roots, which we take to be α1, α3, α6 and α8, not in
Φs. We obtain the following weight nets with their contribution to |Λ| given in brackets:
· 3 · 6 3 0 -3 3 (2), · 21 18 15 12 9 6 3 (1), · 12 · 15 12 9 6 3 (3), · 3 · 15 12 9 6 3 (3), · 12 9 6 3 · 6 3 (3),
· 12 9 6 12 9 6 3 (1), -3 -6 · 15 12 9 6 3 (1), · 12 9 6 3 0 -3 · (3), · 3 · 6 3 · 6 3 (4), · 3 0 -3 12 9 6 3 (1),
-3 -6 · 6 12 9 6 3 (1), · 3 0 -3 3 · 6 3 (2), -3 -6 -9 6 3 · 6 3 (1), · 3 0 -3 -6 · 6 3 (3), · 3 · 6 3 0 -3 · (4),
-3 -6 · 6 3 · 6 3 (2), -3 -6 · -3 12 9 6 3 (1) and -3 -6 · -3 3 · 6 3 (2).
Taking these weight nets together with the nets obtained by negating signs and reversing
coefficients we have |Λ| > 76, so (♦) is satisfied for each X ∈ I.
Consider L(ω1+ω2) for n = 7. From Figure 5.6 and (♦) we can assume that there are
two orthogonal roots not in Φs; we may take these to be α1 and α7. We provide weight
nets below with weights in W.ω3 italicised since these have multiplicity 2 when p 6= 3.
Note that we omit a factor of 1
8
on each coefficient.
81
1× 2 nets: · 18 15 12 9 6 3, · 2 15 12 9 6 3, · 2 -1 12 9 6 3, · 2 -1 -4 9 6 3, · 2 -1 -4 -7 6 3,
· 2 7 12 9 6 3, · 2 7 4 9 6 3, · 2 7 4 1 6 3, · 2 -1 4 9 6 3, · 2 -1 -4 1 6 3, · 2 -1 4 1 6 3
2× 2 nets: · 2 7 4 1 -2 ·, · 2 -1 4 1 -2 ·
1× 3 nets: · 10 15 12 9 6 3, · 10 7 12 9 6 3, · 10 7 4 9 6 3, · 10 7 4 1 6 3
2× 3 nets: · 10 7 4 1 -2 ·
We remark that the 1 × 3 and 2 × 3 weight nets consist of strings of the form µ2 µ1
µ2. Taking these weight nets together with the corresponding ones with weights negated
and coefficients reversed we have |Λ| > 74, 76 or 44 according as p 6= 2, 3, p = 2 or
p = 3. Hence we may assume that p = 3. We see that (♦) holds for X = A4 and larger
subsystems of A7 so we may assume that α1, α3, α5 and α7 do not lie in Φs. The weight
nets (up to negation and reversing coefficients) together with their contribution to the
codimension of the eigenspace in brackets are as follows:
· 18 15 12 9 6 3 (1), -3 -6 · 12 9 6 3 (1), · 10 · 12 9 6 3 (3), · 10 7 4 · 6 3 (3), · 2 · 12 9 6 3 (3),
· 10 7 4 1 -2 · (3), · 2 -1 -4 · 6 3 (3), -3 -6 · 4 · 6 3 (3), · 2 · 4 · 6 3 (4) and · 2 · 4 1 -2 · (4).
Thus |Λ| > 56 and (♦) holds unless X = ∅. If X = ∅ then we can arrange the weights
into cliques to show that d∅ω1+ω2 > 58 > 56 = e
∅
ω1+ω2 ; we use the eight cliques given
below together with the counterparts of the first four which are obtained by reversing the
coefficients and negating.
13 18 15 12 9 6 3, 5 18 15 12 9 6 3, . . . , 5 10 7 4 1 -2 -5 13 10 7 12 9 6 3, . . . , 13 10 7 4 1 -2 -5
5 2 15 12 9 6 3, . . . , 5 2 7 4 1 -2 -5 -3 10 15 12 9 6 3, . . . , -3 10 7 4 1 -2 -5
-3 2 7 4 9 6 3, -3 -6 7 4 9 6 3, -3 -6 -1 4 9 6 3, -3 -6 -1 -4 9 6 3, -3 -6 -1 -4 1 -2 3
-3 -6 -9 4 9 6 3, . . . , -3 -6 -9 4 1 -2 -5 5 2 -1 -4 1 6 3, . . . , -3 -6 -9 -12 -7 -2 3
-3 2 -1 -4 1 -2 -5, . . . , -3 -6 -9 -12 1 -2 -5
Consider L(ω1 + ω2) for n = 6. From Figure 5.6 and using condition (♦), as before,
we can assume that there are two orthogonal roots not in Φs; take these to be α1 and α6.
We provide weight nets below with weights in W.ω3 italicised. Note that we omit a factor
of 1
7
on each coefficient.
82
1× 2 nets: · 15 12 9 6 3, · 1 12 9 6 3 , · 1 -2 9 6 3, · 1 -2 -5 6 3, · 1 5 9 6 3, · 1 5 2 6 3, · 1 -2 2 6 3
2× 2 nets: · 1 5 2 -1 ·, · 1 -2 2 -1 ·
1× 3 nets: · 8 12 9 6 3 , · 8 5 9 6 3, · 8 5 2 6 3
2× 3 net: · 8 5 2 -1 ·
Using these weight nets together with the corresponding ones with weights negated
and coefficients reversed (although note that the net · 1 -2 2 -1 · is its own counterpart)
we have |Λ| > 52, 54 or 32 according as p 6= 2, 3, p = 2 or p = 3. Hence we may assume
that p = 3. We see that (♦) holds for X = A4 and larger subsystems of A6 so we may
assume that α1, α3 and α6 do not lie in Φs. The weight nets are as follows.
2× 1× 1, 1× 2× 1, 1× 1× 2 nets: · 15 12 9 6 3, -3 -6 · -5 6 3, -3 -6 -9 -12 -1 ·,
· 1 -2 -5 6 3, -3 -6 · 9 6 3, -3 -6 -9 -12 -15 ·
1× 2× 2, 2× 1× 2, 2× 2× 1 nets: -3 -6 · -5 -1 ·, · 1 -2 -5 -1 ·, · 1 · 2 6 3
3× 1× 1, 1× 3× 1, 1× 1× 3 nets: · 8 5 2 6 3, -3 -6 · 2 6 3, -3 -6 -9 -12 -8 ·
1× 2× 3, . . . , 3× 2× 1 nets: -3 -6 · -5 -8 ·, -3 -6 · 2 -1 ·, · 1 -2 -5 -8 ·, · 1 · 9 6 3, · 8 5 2 -1 ·, · 8 · 9 6 3
2× 2× 2 net: · 1 · 2 -1 ·
Thus |Λ| > 37 and (♦) holds unless X = ∅, A1 or A21.
Suppose thatX = A21 with simple roots α1 and α6. From the table below, by arranging
clusters into cliques, we have d
A2
1
ω1+ω2 > 39 > 38 = e
A2
1
ω1+ω2 .
(λ, n, X) = (ω1 + ω2, 6, A21)
Cliques Cluster size l Cliques Cluster size l
· 15 12 9 6 3 2 11 · 1 -2 -5 -8 · 6 11
· 8 12 9 6 3 3 -3 -6 -2 -5 -8 · 3
· 8 5 9 6 3 3 -3 -6 -9 -5 -8 · 3
· 8 5 2 6 3 3 -3 -6 -9 -12 -8 · 3
· 8 5 2 -1 · 6 -3 -6 -9 -12 -15 · 2
· 1 12 9 6 3 2 6 · 1 -2 -5 -1 · 4 6
· 1 5 9 6 3 2 -3 -6 -2 -5 -1 · 2
· 1 5 2 6 3 2 -3 -6 -9 -5 -1 · 2
· 1 5 2 -1 · 4 -3 -6 -9 -12 -1 · 2
· 1 -2 9 6 3 2 4 -3 -6 -2 2 6 3 1 1
· 1 -2 2 6 3 2 -3 -6 -9 2 6 3 1
· 1 -2 2 -1 · 4
If X = A1 with simple root α1 then d
A1
ω1+ω2 > 41 > 40 = e
A1
ω1+ω2 from the table below.
83
(λ, n, X) = (ω1 + ω2, 6, A1)
Cliques Cluster size l Cliques Cluster size l Cliques Cluster size l
· 15 12 9 6 3 2 14 · 1 12 9 6 3 2 2 · 1 5 2 -1 -4 2 2
· 8 12 9 6 3 3 · 1 5 9 6 3 2 · 1 -2 2 -1 -4 2
· 8 5 9 6 3 3 · 1 5 2 -1 3 2 2 · 1 -2 -5 -8 3 2 3
· 8 5 2 6 3 3 · 1 -2 2 -1 3 2 -3 -6 -2 -5 -8 3 1
· 8 5 2 -1 3 3 · 1 -2 -5 -8 -4 2 2 -3 -6 -9 -5 -8 3 1
· 8 5 2 -1 -4 3 · 1 -2 -5 -8 -11 2 -3 -6 -9 -12 -8 3 1
· 1 5 2 6 3 2 8 -3 -6 -2 2 6 3 1 4 -3 -6 -2 -5 -8 -4 1 4
· 1 -2 2 6 3 2 -3 -6 -9 2 6 3 1 -3 -6 -9 -5 -8 -4 1
· 1 -2 -5 6 3 2 -3 -6 -9 -5 6 3 1 -3 -6 -9 -12 -8 -4 1
· 1 -2 -5 -1 3 2 -3 -6 -9 -5 -1 3 1 -3 -6 -9 -12 -15 -4 1
· 1 -2 -5 -1 -4 2 -3 -6 -9 -5 -1 -4 1 -3 -6 -9 -12 -15 -11 1
If X = ∅ then we see that d∅ω1+ω2 > 43 > 42 = e
∅
ω1+ω2 from the cliques below, together
with the counterparts of the first three cliques given (obtained by reversing the coefficients
and negating).
11 15 12 9 6 3, 4 15 12 9 6 3, . . . , 4 8 5 2 -1 -4 11 8 12 9 6 3, . . . , 11 8 5 2 -1 -4 -3 8 12 9 6 3, . . . , -3 8 5 2 -1 -4
4 1 5 9 6 3, -3 1 5 9 6 3, . . . , -3 -6 -2 2 -1 -4 -3 -6 -9 2 6 3, . . . , -3 -6 -9 -5 -1 -4 4 1 -2 9 6 3, -3 1 -2 9 6 3
4 1 -2 -5 6 3, . . . , -3 -6 -2 -5 6 3 4 1 12 9 6 3, -3 1 12 9 6 3 4 1 5 2 6 3, . . . , -3 -6 5 2 6 3
Finally, consider L(ω1+ω2) for n = 5. We repeat the same procedure as above. From
Figure 5.6 and (♦) we can take α1, α5 /∈ Φs. We provide weight nets below with weights
in W.ω3 italicised. We have omitted a factor of
1
6
on each coefficient.
1× 2 nets: · 12 9 6 3, · 0 9 6 3 , · 0 -3 6 3, · 0 3 6 3 2× 2 net: · 0 3 0 ·
1× 3 nets: · 6 9 6 3 , · 6 3 6 3 2× 3 net: · 6 3 0 ·
Using these weight nets together with the corresponding ones with weights negated
and coefficients reversed we have |Λ| > 34, 36 or 22 according as p 6= 2, 3, p = 2 or p = 3.
Thus (♦) is satisfied if p 6= 3 and we may assume that p = 3. We see that (♦) holds for
X = A3 and larger subsystems of A5 so we may assume that α1, α3 and α5 do not lie in
Φs. The weight nets are as follows.
1× 1× 2, 1× 2× 1, 2× 1× 1 nets: -3 -6 -9 -12 ·, -3 -6 · 6 3, · 12 9 6 3
1× 2× 3, . . . , 3× 2× 1 nets: -3 -6 · -6 ·, -3 -6 · 0 ·, · 0 -3 -6 ·, · 0 · 6 3, · 6 3 0 ·, · 6 · 6 3
2× 2× 2 net: · 0 · 0 ·
Thus |Λ| > 25 and (♦) holds unless X = ∅, A1 or A21.
84
Suppose thatX = A21 with simple roots α1 and α5. From the table below, by arranging
clusters into cliques, we have d
A2
1
ω1+ω2 > 30 > 28 = e
A2
1
ω1+ω2 .
(λ, n, X) = (ω1 + ω2, 5, A21)
Cliques Cluster size l Cliques Cluster size l Cliques Cluster size l
· 12 9 6 3 2 5 -3 -6 -3 -6 · 3 5 · 6 3 0 · 6 14
· 6 9 6 3 3 -3 -6 -9 -6 · 3 · 0 3 0 · 4
· 6 3 6 3 3 -3 -6 -9 -12 · 2 · 0 -3 0 · 4
· 0 9 6 3 2 2 -3 -6 -3 0 · 2 2 · 0 -3 -6 · 6
· 0 3 6 3 2 -3 -6 -9 0 · 2
· 0 -3 6 3 2 1 -3 -6 3 6 3 1 1
-3 -6 -3 6 3 1 -3 -6 3 0 · 2
If X = A1 with simple root α1 then d
A1
ω1+ω2 > 29 > 28 = e
A1
ω1+ω2 from the table below.
(λ, n, X) = (ω1 + ω2, 5, A1)
Cliques Cluster size l Cliques Cluster size l Cliques Cluster size l
· 12 9 6 3 2 11 · 0 -3 -6 -3 2 4 · 0 3 6 3 2 4
· 6 9 6 3 3 -3 -6 -3 -6 -3 1 -3 -6 3 6 3 1
· 6 3 6 3 3 -3 -6 -9 -6 -3 1 -3 -6 -3 6 3 1
· 6 3 0 3 3 -3 -6 -9 -12 -3 1 -3 -6 -3 0 3 1
· 6 3 0 -3 3 -3 -6 -9 -12 -9 1 -3 -6 -3 0 -3 1
· 0 -3 -6 -9 2 2 · 0 -3 6 3 2 4 · 0 9 6 3 2 4
-3 -6 -3 -6 -9 1 · 0 -3 0 3 2 · 0 3 0 3 2
-3 -6 -9 -6 -9 1 · 0 -3 0 -3 2 · 0 3 0 -3 2
If X = ∅ then d∅ω1+ω2 > 32 > 30 = e
∅
ω1+ω2 from the cliques below.
9 12 9 6 3, 3 12 9 6 3, . . . , 3 6 3 0 -3 3 0 -3 -6 -3, -3 0 -3 -6 -3, . . . , -3 -6 -9 -12 -9
9 6 9 6 3, . . . , 9 6 3 0 -3 3 0 -3 -6 -9, . . . , -3 -6 -9 -6 -9 3 0 9 6 3, . . . , 3 0 3 0 -3 3 0 -3 0 -3, . . . , -3 -6 -9 0 -3
-3 6 9 6 3, . . . , -3 0 3 0 -3 3 0 -3 0 3, . . . , -3 -6 -9 -6 3 3 0 -3 6 3, . . . , -3 -6 -3 6 3
The lemma follows for L(3ω1) with n ∈ [5, 8] by Premet’s theorem.
5.5 Groups of low rank
In this section we shall consider in turn the groups of type An for n ∈ [1, 4]. The
calculations for each irreducible G-module are similar to those when we considered groups
85
of higher rank, so we can withhold the full details of calculations for most of them. It will
suffice to exhibit the results of the calculations in tables. As in the previous section, on
the left hand side are tables with information on each module pertaining to weight orbits
and multiplicities. This information is taken from [18]. On the right hand side are tables
detailing the weight strings that occur and their number. From this we can calculate a
lower bound for the codimension of the eigenspace.
5.5.1 Case I: n = 4
We begin with the module with highest weight ω2 + ω3; we shall give full details of the
calculations performed.
Lemma 5.15. Let G be as above acting on the irreducible module L(ω2 + ω3). Then (†)
holds when p 6= 3 and (♦) is satisfied for all X ∈ I when p = 3.
Proof. From Lu¨beck’s tables we see that there are three orbits on weights, with dominant
weights ω2 + ω3, ω1 + ω4 and 0; we label these orbits with i = 2, 1, 0 respectively.
Given µ ∈ W.(ω2+ω3), we have 〈µ, α〉 = 0, 1 and 2 for four roots α in each case. Since
|W.(ω2 + ω3)| =
|W (A4)|
|W (A2
1
)|
= 30, we see that given α ∈ Φ(A4) there are
30.4
20
= 6 weights
µ ∈ W.(ω2+ω3) orthogonal to α, there are 6 weights with weight string µ, µ−α and there
are 6 weights with weight string µ, µ − α, µ − 2α. Given µ′ ∈ W.(ω1 + ω4), we calculate
that 〈µ′, α〉 = 0 and 〈µ′, α〉 = 1 for 6 roots α in each case, and 〈µ′, α〉 = 2 for 1 root
α. We have |W.(ω1 + ω4)| =
|W (A4)|
|W (A2)|
= 20. Given α ∈ Φ(A4) there are
20.6
20
= 6 weights
µ′ ∈W.(ω1+ω4) orthogonal to α, there are 6 weights µ
′ ∈W.(ω1+ω4) with weight string
µ1, µ1−α and there is 1 weight µ
′ ∈W.(ω1+ω4) with weight string µ
′, µ′−α, µ′−2α. We
note that the weights in W.(ω1+ω4) orthogonal to α are contained in the weight string of
length three when 〈µ, α〉 = 2. Taking multiplicities into account, from each weight string
we find the minimal value of |Λ| (or equivalently, the maximum value of Π(ω2+ω3)\ |Λ|).
The calculations detailed are displayed in Figure 5.7.
When p 6= 3 we find that (†) holds. If p = 3 then |Λ| > 19 and we see that (♦) is
satisfied unless X = ∅. The five cliques given, together with their negative counterparts,
86
i ω |W.ω| mω
p 6= 2, 3 p = 2 p = 3
2 ω2 + ω3 30 1 1 1
1 ω1 + ω4 20 2 2 1
0 0 1 5 4 1
Weight No. of l
strings strings p 6= 2, 3 p = 2 p = 3
µ2 6
µ2 µ2 6 6 6 6
µ2 µ1 µ2 6 12 12 6
µ1 µ1 6 12 12 6
µ1 µ0 µ1 1 4 4 1
Lower bound on |Λ| 34 34 19
Figure 5.7: λ = ω2 + ω3
show that d∅ω2+ω3 > 2.12 = 24 > 20 = e
∅
ω2+ω3 .
1 2 2 1 0 1 1 1 1 0 1 1 1 1 0 1 1 2 1 1
1 1 2 1 0 0 1 1 1 0 1 0 1 0 0 1 1 2 1 0
1 1 1 1 0 0 0 1 1 0 0 0 1 0 -1 0 1 1 0 0
1 1 1 0 0 1 0 0
In the next lemma we deal with irreducible modules parameterised by highest weights
with level 4. We shall see that in low rank cases it is often necessary to list all of the
weights that occur for a given module.
Lemma 5.16. If G acts on L(2ω1 + ω2) (p > 2) then (†) holds, and if it acts on L(4ω1)
(p > 3), L(ω1 + ω3) or L(2ω2) (p > 2) then (♦) is satisfied for all X ∈ I.
Proof. The calculations involving weight strings for L(2ω1+ω2) and L(4ω1) are contained
in Figures 5.8 and 5.9.
i ω |W (ω)| mω
p 6= 2
4 2ω1 + ω2 20 1
3 2ω2 10 1
2 ω1 + ω3 30 2
1 ω4 5 3
Weight No. of l
strings strings p 6= 2
µ4 6
µ4 µ4 3 3
µ4 µ3 µ4 1 1
µ4 µ2 µ2 µ4 3 9
µ3 3
µ3 µ2 µ3 3 6
µ2 3
µ2 µ2 6 12
µ2 µ1 µ2 3 9
µ1 µ1 1 3
Lower bound on |Λ| 43
Figure 5.8: λ = 2ω1 + ω2
From the Figures we see that (†) is satisfied for L(2ω1 + ω2) and (♦) holds for all
X ∈ I for L(4ω1).
87
i ω |W.ω| mω
p 6= 2, 3
5 4ω1 5 1
4 2ω1 + ω2 20 1
3 2ω2 10 1
2 ω1 + ω3 30 1
1 ω4 5 1
Weight No. of l
strings strings p 6= 2, 3
µ5 3
µ5 µ4 µ3 µ4 µ5 1 2
µ4 6
µ4 µ4 3 3
µ4 µ2 µ2 µ4 3 6
µ3 3
µ3 µ2 µ3 3 3
µ2 3
µ2 µ2 6 6
µ2 µ1 µ2 3 3
µ1 µ1 1 1
Lower bound on |Λ| 24
Figure 5.9: λ = 4ω1
Now consider L(ω1 + ω3) and L(2ω2); the usual calculations involving weight strings
are contained in Figures 5.10 and 5.11. We note that although ω1 + ω3 < 2ω2 it will not
suffice to consider only ω1+ω3 and use Premet’s theorem. The multiplicity of each weight
in W.(ω4) for L(ω1 + ω3) is higher than that for L(2ω2).
i ω |W.ω| mω
p 6= 2 p = 2
2 ω1 + ω3 30 1 1
1 ω4 5 3 2
Weight No. of l
strings strings p 6= 2 p = 2
µ2 6
µ2 µ2 9 9 9
µ2 µ1 µ2 3 6 6
µ1 µ1 1 3 2
Lower bound on |Λ| 18 17
Figure 5.10: λ = ω1 + ω3
i ω |W.ω| mω
p 6= 2, 3 p = 3
3 2ω2 10 1 1
2 ω1 + ω3 30 1 1
1 ω4 5 2 1
Weight No. of l
strings strings p 6= 2, 3 p = 3
µ3 4
µ3 µ2 µ3 3 3 3
µ2 3
µ2 µ2 9 9 9
µ2 µ1 µ2 3 6 3
µ1 µ1 1 2 1
Lower bound on |Λ| 20 16
Figure 5.11: λ = 2ω2
Let us consider the module L(2ω2); we first assume p = 3. We see that |Λ| > 16 and
(♦) is satisfied if |Φ(X)| > 20 − 16 = 4, so the centraliser types requiring consideration
are X = ∅, A1 or A21. The weight nets for X = A
2
1 are given in Figure 5.12.
Suppose X = A21 with simple roots α1, α4; we have e
A2
1
2ω2
= 16. The weight net dia-
gram (Figure 5.12) displays the clusters of weights. The weights in italics and bold lie
88
6 12 8 4
4¯ 8¯ 12 6¯ 4¯ 8¯ 7¯ 6¯ 4¯ 8¯ 7¯ 1¯ 4¯ 8¯ 2¯ 6¯ 4¯ 8¯ 2¯ 1¯ 4¯ 8¯ 2¯ 4
6 7 8 4 6 7 3 1¯ 6 7 3 4
1 7 8 4 1 7 3 1¯ 1 7 3 4
1 3¯ 7¯ 6 1 3¯ 7¯ 1¯ 1 3¯ 3 1¯ 1 3¯ 3 4 1 3¯ 2¯ 6¯ 1 3¯ 2¯ 1¯ 1 3¯ 2¯ 4
4¯ 3¯ 7¯ 6¯ 4¯ 3¯ 7¯ 1¯ 4¯ 3¯ 3 1¯ 4¯ 3¯ 3 4 4¯ 3¯ 2¯ 6¯ 4¯ 3¯ 2¯ 1¯ 4¯ 3¯ 2¯ 4
6 2 8 4 6 2 3 1 6 2 3 4 6 2 2¯ 6¯ 6 2 2¯ 1¯ 6 2 2¯ 4
1 2 8 4 1 2 3 1¯ 1 2 3 4 1 2 2¯ 6¯ 1 2 2¯ 1¯ 1 2 2¯ 4
4¯ 2 8 4 4¯ 2 3 1¯ 4¯ 2 3 4 4¯ 2 2¯ 6¯ 4¯ 2 2¯ 1¯ 4¯ 2 2¯ 4
Figure 5.12: Weight nets for λ = 2ω2 and X = A
2
1
in W.(2ω2) and W.ω4 respectively, and we set x = −x for a coefficient x of a weight.
Suppose we have any weight in the cluster · 2 -2 · in Π(V ) \ Λ, then the clusters
· -3 -2 ·, · 2 3 ·, · 7 3 ·, · -3 -7 ·
lie in Λ by the adjacency principle, hence d
A2
1
2ω2
> 20. So assume that this is not the case.
Suppose we have a weight from the cluster · 2 3 · in Π(V ) \Λ. Then we cannot have any
weight in the clusters · -3 3 ·, · 2 8 ·, · 7 3 · in Π(V ) \ Λ, so d
A2
1
2ω2
> 9 + 11 = 20. Hence,
we must have the cluster · 2 3 · in Λ and by symmetry, the cluster · -3 -2 · must lie in Λ.
We have now eliminated three clusters of size 9, 6 and 6, so we are done.
Suppose X = A1 with simple root α4; we have e
A1
2ω2
= 18. We can have at most one of
the clusters in Π(V ) \ Λ from each of the following cliques:
1 -3 -7 · 1 2 -2 · 6 7 3 · 6 2 3 · -4 2 3 · 1 2 3 · 6 7 8 4
-4 -3 -7 · 1 -3 -2 · 1 7 3 · 6 2 -2 · -4 2 -2 · 1 -3 3 · 1 7 8 4
-4 -8 -7 · -4 -3 -2 · -4 -3 3 · 1 2 8 4
If we ignore any weights in W.(2ω2) we obtain d
A1
2ω2
> 4 + 6 + 2 + 1 + 1 + 4 + 2 = 20
and we are done.
Suppose that X = ∅; we have e∅2ω2 = 20. We can form the following seven cliques:
6 7 8 4 6 2 3 4 -4 2 3 4 1 2 3 4 1 2 -2 4 -4 -8 -2 -1 1 2 -2 -6
1 7 8 4 6 2 3 -1 -4 2 3 -1 1 2 3 -1 1 -3 -2 4 -4 -8 -7 -1 1 -3 -2 -6
1 2 8 4 6 2 -2 -1 -4 2 -2 -1 1 2 -2 -1 -4 -3 -2 4 -4 -8 -7 -6 -4 -3 -2 -6
1 -3 -2 -1
-4 -3 -2 -1
We can see from the weight net above that the 12 remaining weights in the orbit of
ω1 + ω3 (those of the form · -3 -7 ·, · 7 3 · and · -3 3 ·) can be taken in pairs, with each
weight in a pair differing by a root. Here at least one of the two weights in each pair of
weights is in Λ implying that d∅2ω2 > 22.
When p 6= 2, 3, the 5 weights in the orbit W.(ω4) have multiplicity 2 and we know
that |Λ| > 20 whence the only centraliser type requiring further consideration is X = ∅.
89
Clearly the same calculation as above for characteristic 3 deals with this possibility.
Finally, consider the module L(ω1+ω3). We find that |Λ| > 18 or 17 according as p 6= 2
or p = 2. The condition (♦) is satisfied unless X = ∅ or A1 for all characteristics. We
can use the exact same analysis as above for both of these possibilities since we avoided
using weights from the orbit W.(2ω2).
The next lemma deals with weights with level 3 and will complete the investigation
for groups of type A4.
Lemma 5.17. If G acts on L(2ω1+ω4) (p > 2) then (†) holds, and if it acts on L(ω1+ω2)
or L(3ω1) (p > 3) then (♦) is satisfied for all X ∈ I.
Proof. We perform the usual calculations for these modules. Considering weight strings
for L(ω1 + ω2) we obtain Figure 5.13.
i ω |W.ω| mω
p 6= 3 p = 3
2 ω1 + ω2 20 1 1
1 ω3 10 2 1
Weight No. of l
strings strings p 6= 3 p = 3
µ2 6
µ2 µ2 4 4 4
µ2 µ1 µ2 3 6 3
µ1 1
µ1 µ1 3 6 3
Lower bound on |Λ| 16 10
Figure 5.13: λ = ω1 + ω2
When p = 3, (♦) is satisfied provided that |Φ(X)| > 20 − 10 = 10 and so we
need to consider the centraliser types X = ∅, A1, A21, A2 and A2A1; when p 6= 3
only the first three of these possibilities need to be examined. Indeed, it will suf-
fice to consider the case p = 3. We shall omit a factor of 1
5
on each coefficient of a
weight. First suppose that X = A2A1 with simple roots α1, α3 and α4; there are four
clusters in this case. We list these with cluster size in brackets after each of them:
· 9 6 3 (2), · 4 · · (9), · -1 · · (12) and -3 -6 · · (7).
Assuming that the largest cluster · -1 · · is in Π(V )\Λ means that dA2A1ω1+ω2 > 16 by the ad-
jacency principle; this exceeds eA2A1ω1+ω2 = 12. Otherwise, applying the adjacency principle
to the clusters · 9 6 3 and · 4 · · we see that dA2A1ω1+ω2 > 12 + 2 = 14, so (♦) is satisfied.
90
If X = A2 with simple roots α3 and α4 then the clusters are
7 9 6 3 (1), 2 9 6 3 (1), 7 4 · · (3), 2 4 · · (3), -3 4 · · (3),
2 -1 · · (6), -3 -1 · · (6) and -3 -6 · · (7).
We can form three cliques with the clusters of size one, the clusters of sizes six and seven
and two of the clusters of size three. Hence dA2ω1+ω2 > 1 + 12 + 3 = 16 > 14 = e
A2
ω1+ω2 .
If X = A21 with simple roots α1 and α4 then e
A2
1
ω1+ω2 = 16 and there are nine clusters
· 9 6 3 (2), · 4 6 3 (3), · 4 1 · (6), · -1 6 3 (2), · -1 1 · (4),
· -1 -4 · (6), -3 -6 1 · (2), -3 -6 -4 · (3) and -3 -6 -9 · (2).
If the cluster · 4 1 · lies in Π(V ) \ Λ then the clusters
· 9 6 3, · 4 6 3, · -1 1 · and · -1 -4 ·
contribute to the codimension of the eigenspace. The remaining four clusters form two
cliques so we have d
A2
1
ω1+ω2 > 19. Thus we may assume that · 4 1 · lies in Λ and we may
assume the same for · -1 -4 · and we have d
A2
1
ω1+ω2 > 18 by forming cliques with six of the
remaining clusters, namely · 9 6 3 and · 4 6 3, · -1 6 3 and · -1 1 · and finally -3 -6 -4 ·
and -3 -6 -9 ·.
If X = A1 with Φs = {±α4} then the following cliques show that d
A1
ω1+ω2 > 19 > 18 =
eA1ω1+ω2 .
Clique Cluster size l Clique Cluster size l
7 9 6 3 1 2 2 4 6 3 1 1
7 4 6 3 1 -3 4 6 3 1
7 4 1 · 2 2 -1 6 3 1 1
2 9 6 3 1 3 -3 -1 6 3 1
2 4 1 · 2 2 -1 -4 · 3 8
-3 4 1 · 2 -3 -1 -4 · 3
2 -1 1 · 2 4 -3 -6 -4 · 3
-3 -1 1 · 2 -3 -6 -9 · 2
-3 -6 1 · 2
Similarly, if X = ∅ then the following cliques show that d∅ω1+ω2 > 22 > 20 = e
∅
ω1+ω2 .
7 4 6 3 7 9 6 3 -3 -1 1 3 2 -1 -4 3 2 -1 -4 -7 7 4 1 3 -3 4 6 3 -3 -1 6 3
2 4 6 3 2 9 6 3 -3 -1 1 -2 -3 -1 -4 3 -3 -1 -4 -7 7 4 1 -2 -3 4 1 3 -3 -6 1 3
2 -1 6 3 2 4 1 3 -3 -1 -4 -2 -3 -6 -4 3 -3 -6 -4 -7 2 -1 -4 -2 -3 4 1 -2 -3 -6 1 -2
2 -1 1 3 2 4 1 -2 -3 -6 -4 -2 -3 -6 -9 -2 -3 -6 -9 -7
2 -1 1 -2
Now we shall consider the module L(3ω1); the calculations are shown in Figure 5.14.
91
i ω |W.ω| mω
p 6= 2, 3
3 3ω1 5 1
2 ω1 + ω2 20 1
1 ω3 10 1
Weight No. of l
strings strings p 6= 2, 3
µ3 3
µ3 µ2 µ2 µ3 1 2
µ2 6
µ2 µ2 3 3
µ2 µ1 µ2 3 3
µ1 1
µ1 µ1 3 3
Lower bound on |Λ| 11
Figure 5.14: λ = 3ω1
If the centraliser type satisfies 9 < |Φ(X)| then we are done; otherwise the centraliser
types requiring further consideration are the same as for L(ω1 + ω2) when p = 3. Since
ω1 + ω2 < 3ω1, the same analysis for L(ω1 + ω2) when p = 3 above deals with all the
centraliser types requiring consideration.
Finally, considering L(2ω1+ ω4) we see in Figure 5.15 that (†) holds for all character-
istics.
i ω |W.ω| mω
p 6= 2, 3 p = 3
3 2ω1 + ω4 20 1 1
2 ω2 + ω4 30 1 1
1 ω1 5 4 3
Weight No. of l
strings strings p 6= 2, 3 p = 3
µ3 6
µ3 µ3 3 3 3
µ3 µ2 µ3 3 3 3
µ3 µ1 µ1 µ3 1 5 4
µ2 3
µ2 µ2 9 9 9
µ2 µ1 µ2 3 6 6
Lower bound on |Λ| 26 25
Figure 5.15: λ = 2ω1 + ω4
5.5.2 Case II: n = 3
There are six modules to consider here. We shall start with the four modules of highest
dimension.
Lemma 5.18. If G acts on L(2ω1+ω2) (p > 2) then (†) holds, and if it acts on L(2ω1+ω3)
(p > 2) or L(4ω1) (p > 3) then (♦) is satisfied for each X ∈ I.
Proof. It suffices to perform calculations involving weight strings. These are detailed for
each module in Figures 5.16-5.18. Considering L(2ω1+ω2) we see that |Λ| > dimG so (†)
92
is satisfied and considering L(2ω1+ω3) and L(4ω1) we see in each case that |Λ| > |Φ(G)|
so that (♦) is satisfied for all centraliser types.
i ω |W.ω| mω
p 6= 2, 5 p = 5
3 2ω1 + ω3 12 1 1
2 ω2 + ω3 12 1 1
1 ω1 4 3 2
Weight No. of l
strings strings p 6= 2, 5 p = 5
µ3 2
µ3 µ3 2 2 2
µ3 µ2 µ3 2 2 2
µ3 µ1 µ1 µ3 1 4 3
µ2 µ2 3 3 3
µ2 µ1 µ2 2 4 4
Lower bound on |Λ| 15 14
Figure 5.16: λ = 2ω1 + ω3
i ω W.ω mω
p 6= 2, 3
4 4ω1 4 1
3 2ω1 + ω2 12 1
2 2ω2 6 1
1 ω1 + ω3 12 1
0 0 1 1
Weight No. of l
strings strings p 6= 2, 3
µ4 2
µ4 µ3 µ2 µ3 µ4 1 2
µ3 2
µ3 µ3 2 2
µ3 µ1 µ1 µ3 2 4
µ2 1
µ2 µ1 µ2 2 2
µ1 µ1 2 2
µ1 µ0 µ1 1 1
Lower bound on |Λ| 13
Figure 5.17: λ = 4ω1
i ω |W.ω| mω
p 6= 2
3 2ω1 + ω2 12 1
2 2ω2 6 1
1 ω1 + ω3 12 2
0 0 1 3
Weight No. of l
strings strings p 6= 2
µ3 2
µ3 µ3 2 2
µ3 µ2 µ3 1 1
µ3 µ1 µ1 µ3 2 6
µ2 1
µ2 µ1 µ2 2 4
µ1 µ1 2 4
µ1 µ0 µ1 1 3
Lower bound on |Λ| 20
Figure 5.18: λ = 2ω1 + ω2
Lemma 5.19. If G acts on the irreducible module L(3ω1) (p > 3) then (♦) is satisfied
for all X ∈ I.
Proof. From Figure 5.19 we see that |Λ| > 7 for p 6= 2, 3.
If |Φ(X)| > 5 then (♦) is satisfied so X = ∅, A1, A21 are the centraliser types requiring
further consideration. First consider the case X = A21 and note that e
A2
1
3ω1
= 8. There are
93
i ω |W.ω| mω
p 6= 2, 3
3 3ω1 4 1
2 ω1 + ω2 12 1
1 ω3 4 1
Weight No. of l
strings strings p 6= 2, 3
µ3 2
µ3 µ2 µ2 µ3 1 2
µ2 2
µ2 µ2 2 2
µ2 µ1 µ2 2 2
µ1 µ1 1 1
Lower bound on |Λ| 7
Figure 5.19: λ = 3ω1
four clusters of the form · 6 3 (4), · 2 · (6), · -2 · (6) and -3 -6 · (4), where the number in
brackets indicates the size of the cluster and we omit a factor of 1
4
on each coefficient. If
a cluster of size 6 is in Π(V ) \ Λ then the adjacency principle shows that d
A2
1
3ω1
> 10. So
no cluster of size 6 can be present and we are done.
If X = A1 with simple root α3 then there are 10 clusters. Note that e
A1
3ω1
= 10. If
the cluster of size 4 is in Π(V ) \ Λ then both clusters of size 3 must be in Λ and the
other clusters can be arranged in cliques as shown below so dA13ω1 > 11. Assuming that the
cluster of size 4 is in Λ, the clusters of size 3 form a clique and together with the other
cliques we find dA13ω1 > 12.
-3 -6 3 1 -2 3 -3 -2 3 1 6 3 9 6 3 5 2 3
-3 -6 -1 1 -2 -1 -3 -2 -1 5 2 -1
-3 -6 -5 1 -2 -5 -3 -2 -5 -3 6 3 5 6 3
-3 -6 -9 1 2 3
3 2 3 1 2 -1
-3 2 -1
If X = ∅ the following cliques show that d∅3ω1 > 15 > 12 = e
∅
3ω1
.
9 6 3 1 6 3 1 2 3 1 -2 3 1 -2 -5
5 6 3 -3 6 3 1 2 -1 -3 -2 3 -3 -2 -5
5 2 3 -3 2 3 1 -2 -1 -3 -6 3 -3 -6 -5
5 2 -1 -3 2 -1 -3 -2 -1 -3 -6 -1 -3 -6 -9
The final two irreducible modules have dimensions very close to dimG and detailed
consideration of the weights is required.
Lemma 5.20. If G acts on the irreducible module V = L(2ω2) (p > 2) then (♦) is
satisfied for all X ∈ I \ {A21}.
Proof. As detailed in Figure 5.20, when p 6= 2, 3 we have |Λ| > 8 and when p = 3 we have
|Λ| > 7. In both cases, (♦) holds unless the centraliser types are X = ∅, A1 or A21.
94
i ω |W.ω| mω
p 6= 2, 3 p = 3
2 2ω2 6 1 1
1 ω1 + ω3 12 1 1
0 0 1 2 1
Weight No. of l
strings strings p 6= 2, 3 p = 3
µ2 2
µ2 µ1 µ2 2 2 2
µ1 µ1 4 4 4
µ1 µ0 µ1 1 2 1
Lower bound on |Λ| 8 7
Figure 5.20: λ = 2ω2
First suppose that X = ∅ in which case e∅2ω2 = 12. Suppose for any characteristic
that the weight 0 is in the eigenspace. This rules out any root lying in the eigenspace,
i.e., the twelve weights in W.(ω1 + ω3). If any of the remaining six weights are in Λ then
we are done. So let us assume that the weights
1 2 1, -1 -2 -1, 1 0 1, -1 0 -1, 1 0 -1 and -1 0 1
are also in the eigenspace. However, this is not possible since X = ∅ so α1 + α2 /∈ Φs.
Thus the weight 0 cannot lie in the eigenspace. We arrange all of the other weights in
cliques as shown below.
1 2 1 -1 0 1 1 0 1 1 0 0 0 1 1 0 0 -1 0 -1 -1
1 1 1 -1 -1 0 0 0 1 1 0 -1 0 1 0 -1 0 -1 -1 -1 -1
1 1 0 0 -1 0 -1 0 0 -1 -2 -1
When p 6= 2, 3 this shows that d∅2ω2 > 13. When p = 3 we need to argue further
as d∅2ω2 > 12. We shall show that all weights in the fifth clique lie in Λ, raising the
codimension by one as required. If the weight 0 1 0 is in Π(V ) \ Λ, then we cannot have
any weight in the sixth clique in Π(V ) \ Λ, so d∅2ω2 > 13. Similarly if the weight -1 0 0
is in Π(V ) \ Λ, then we cannot have any weight in the second clique. If the weight 0 1 1
is in Π(V ) \ Λ then considering the first clique the weights 1 2 1 and 1 1 1 cannot lie in
Π(V ) \ Λ. If the weight 1 1 0 is in Λ then we are done, and if it is not then it rules out
both weights in the fourth clique.
Now suppose that X = A1 with simple root α1; note that e
A1
2ω2
= 10. Arrange the
weights into clusters as shown with weights ofW.(2ω2) given in italics and the zero weight
given in bold.
1 2 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 -1 0 -1 0 0 -1 -1 -1 -2 -1
0 1 1 0 1 0 0 0 1 0 0 0 0 0 -1 -1 -1 0 -1 -1 -1
-1 0 1 -1 0 0 -1 0 -1
First suppose that the weights in the fifth cluster are in Π(V ) \ Λ for all possible
95
characteristics. Then none of the other clusters containing weights in W.(ω1 + ω3) can
be in Π(V ) \ Λ. Thus dA12ω2 > 14 and we can assume that the fifth cluster does not lie in
Π(V ) \ Λ. When p 6= 2, 3 suppose that the weights in the fourth cluster are in Π(V ) \ Λ.
This means that the second and seventh clusters cannot be in Π(V )\Λ. We can have one
of the first and third clusters and one of the sixth and eighth clusters lying in Π(V ) \ Λ.
Thus dA12ω2 > 4+2+2+1+2 = 11 as required. Using the same analysis when p = 3 we only
obtain dA12ω2 > 10. However, we cannot have both the third and sixth clusters in Π(V )\Λ.
Therefore for all p we may assume that the fourth cluster is in Λ (and by symmetry we
may assume the same for the sixth cluster). The first, second and third clusters form a
clique as do the seventh, eighth and ninth. Thus dA12ω2 > 15 or 16 depending on whether
p = 3 or p 6= 2, 3.
Suppose in the previous Lemma that X = A21 with simple roots α1, α3. There are five
cliques: · 2 · and · -2 · each containing one weight, · 1 · and · -1 · with four weights each
and · 0 · containing nine weights. If we can show that the weights in the largest clique
are in Λ then we are done as e
A2
1
2ω2
= 8 here. If the clique · 0 · lies in Π(V ) \ Λ then both
cliques of size four are in Λ. We are done unless both 1 2 1 and -1 -2 -1 lie in Π(V ) \ Λ
which would imply that 2α2(s) = 1, i.e., α2(s) = −1 for any s ∈ SA2
1
. Thus for scalar
multiples λs of the diagonal matrix s = diag(1, 1,−1,−1) we have dimVγ(s) = 11 or 12
depending on whether p = 3 or p 6= 2, 3. In this case our methods are insufficient.
Lemma 5.21. If G acts on the irreducible module L(ω1 + ω2) for p 6= 3 then (♦) holds
for all X ∈ I.
Proof. From Figure 5.21 we see that (♦) is satisfied provided |Φ(X)| > 3.
i ω |W.ω| mω
p 6= 3 p = 3
2 ω1 + ω2 12 1 1
1 ω3 4 2 1
Weight No. of l
strings strings p 6= 3 p = 3
µ2 2
µ2 µ2 3 3 3
µ2 µ1 µ2 2 4 2
µ1 µ1 1 2 1
Lower bound on |Λ| 9 6
Figure 5.21: λ = ω1 + ω2
96
Considering the case X = ∅ we obtain the following cliques (with weights in W.ω3
written in italics and a factor of 1
4
omitted).
1 2 3 5 6 3 5 2 3 -3 2 3 -3 2 -1 1 -2 -5 1 -2 3
1 2 -1 1 6 3 5 2 -1 -3 -2 3 -3 -2 -5 -3 -6 -5 -3 -6 -1
1 -2 -1
-3 -2 -1
Thus d∅ω1+ω2 > 12 = e
∅
ω1+ω2 . Suppose that 123 is in Π(V ) \ Λ. This excludes any
weight in the second and fourth clique so d∅ω1+ω2 > 14. Repeating for the other weights
in W.ω3 we reach the same conclusion and if we assume that all weights in the first clique
are in Λ we are done.
When X = A1 with simple root α4 we arrange the weights into clusters. There are
three clusters containing weights in W.ω3. Assuming that each of these in turn lies in
Π(V ) \Λ we find that dA1ω1+ω2 > 12 > 10 = e
A1
ω1+ω2 as required. If none of these clusters lie
in Π(V ) \ Λ we reach the same conclusion.
We note that L(ω1 + ω2) for n = p = 3 is a small module since it has dimension 16.
5.5.3 Case III: n = 2
There are five irreducible modules in this subsection. The main difficulty occurs when
the dimension of the module being considered is close to that of G.
Lemma 5.22. If G acts on the irreducible modules L(2ω1+2ω2) (p > 2) and L(3ω1+ω2)
(p > 3) then, for both modules, (†) holds for p 6= 5 and (♦) is satisfied for each X ∈ I
otherwise.
Proof. For these modules it suffices just to perform the initial calculations (as shown in
the Figures 5.22 and 5.23) to show that (†) holds, or that (♦) holds for all X ∈ I.
i ω |W.ω| mω
p 6= 2, 5 p = 5
4 2ω1 + 2ω2 6 1 1
3 3ω1 3 1 1
2 3ω2 3 1 1
1 ω1 + ω2 6 2 1
0 0 1 3 1
Weight No. of l
strings strings p 6= 2, 5 p = 5
µ4 µ3 µ4 1 1 1
µ4 µ2 µ4 1 1 1
µ4 µ1 µ0 µ1 µ4 1 4 2
µ3 µ1 µ1 µ3 1 3 2
µ2 µ1 µ1 µ2 1 3 2
Lower bound on |Λ| 12 8
Figure 5.22: λ = 2ω1 + 2ω2
97
i ω |W.ω| mω
p 6= 3, 5 p = 5
4 3ω1 + ω2 6 1 1
3 ω1 + 2ω2 6 1 1
2 2ω1 3 2 1
1 ω2 3 2 1
Weight No. of l
strings strings p 6= 3, 5 p = 5
µ4 µ4 1 1 1
µ4 µ3 µ3 µ4 1 2 2
µ4 µ2 µ1 µ2 µ4 1 4 2
µ3 µ2 µ3 1 2 1
µ3 µ1 µ1 µ3 1 3 2
Lower bound on |Λ| 12 8
Figure 5.23: λ = 3ω1 + ω2
Lemma 5.23. If G acts on the irreducible modules L(2ω1 + ω2) (p > 2) and L(4ω1)
(p > 3) then in both cases (♦) is satisfied for all X ∈ I.
Proof. From Figures 5.24 and 5.25 we see for both modules that we only need to deal
with the possibility that the centraliser has type X = ∅. This can be done easily.
i ω |W.ω| mω
p 6= 2
3 2ω1 + ω2 6 1
2 2ω2 3 1
1 ω1 3 2
Weight No. of l
strings strings p 6= 2
µ3 µ3 1 1
µ3 µ2 µ3 1 1
µ3 µ1 µ1 µ3 1 2
µ2 µ1 µ2 1 1
Lower bound on |Λ| 5
Figure 5.24: λ = 2ω1 + ω2
i ω |W.ω| mω
p 6= 2, 3
4 4ω1 3 1
3 2ω1 + ω2 6 1
2 2ω2 3 1
1 ω1 3 1
Weight No. of l
strings strings p 6= 2, 3
µ4 1
µ4 µ3 µ2 µ3 µ4 1 2
µ3 µ3 1 1
µ3 µ1 µ1 µ3 1 2
µ2 µ1 µ2 1 1
Lower bound on |Λ| 6
Figure 5.25: λ = 4ω1
We are unable to draw the same conclusion about the action of SL3(K) on L(3ω1).
Here the dimension of the module may be the same as the dimension of the closure of the
set of vectors which are eigenvectors for some non-central semisimple element of G and
some eigenvalue.
5.5.4 Case IV: n = 1
It is relatively straightforward in this case to describe the weights which occur; for details
see [13, §7.2]. The SL2(K)-module L(mω1) with m ∈ [0, p − 1] has dimension m + 1
98
and weights mω1, (m − 2)ω1, . . . ,−mω1. We see from Lemma 5.5 that we need only be
concerned with modules having dimensions 6 and 7.
Lemma 5.24. If G = SL2(K) acts on the irreducible modules L(rω1) for r ∈ [5, 6] with
p > 5 then (♦) is satisfied for all X ∈ I.
Proof. Consider the module L(6ω1). The weights are 6ω1, 4ω1, 2ω1, 0, −2ω1, −4ω1 and
−6ω1. By the adjacency principle we have, for any γ ∈ K
∗ and s ∈ Gss \ Z, that |Λ| > 3
and so (♦) is satisfied for all X ∈ I. The same is true for the module L(5ω1).
99
Chapter 6
Groups of type Bn
In this chapter we shall assume that G is a simple simply connected algebraic group of
type Bn with n > 2 defined over an algebraically closed field K and V = L(λ) is an
irreducible G-module with p-restricted highest weight λ. It will be sensible to treat the
cases p 6= 2 and p = 2 separately since the conclusion of Premet’s theorem does not hold
in the latter case.
We shall prove the following result.
Theorem 6.1. Let G = Spin2n+1(K) act on V = L(λ). If dimV 6 dimG + 2 then
dimE = dimV with the possible exceptions of L(ω1 + ω2) for n = 2 with p = 5 and
L(ωn) for n ∈ [5, 6]; if instead dimV > dimG + 2 then dimE < dimV with the possible
exception of L(2ω1) for n > 2 with p 6= 2.
This theorem is a consequence of the results to follow in later sections.
6.1 Initial survey
Assume that p 6= 2. We shall deal with the case p = 2 later in Section 6.2. Consider
µ 6 λ where µ =
∑n
i=1 aiωi is a dominant weight. We shall begin by obtaining conditions
on the coefficients ai in order to show that (†) is satisfied for n large enough. This will
allow us later to list modules which will require further consideration. Recall that there is
100
a short root outside the root system of the centraliser of a non-central semisimple element
and in calculations to follow we shall be considering short roots only: see Section 4.3.
It is useful to note that for 1 < k 6= n we have ωk−1 < ωk; this is clear since each
fundamental weight is of the form ωi = α1 + 2α2 + · · · + (i − 1)αi−1 + i (αi + · · ·+ αn)
for i ∈ [1, n − 1]. Also we have ωn−1 < 2ωn since ωn =
1
2
(α1 + 2α2 + · · ·+ nαn). In this
section and the next, to avoid any confusion, we shall occasionally use the notation Λλ
rather than just Λ to emphasise that we are considering the module L(λ).
Proposition 6.2. Suppose that µ =
∑n
i=1 aiωi 6 λ is a dominant weight, and at least
one of the following conditions holds:
(i) n > 6 and ak 6= 0 for some k ∈ [4, n− 1];
(ii) n > 9 and an 6= 0;
(iii) n > 5 and ai, an 6= 0 for some i ∈ [1, 3];
(iv) n = 5 and ai, a4 6= 0 for some i ∈ [1, 3];
(v) n > 5 and ai, a3 6= 0 for some i ∈ [1, 2].
Then (†) holds.
Proof. We shall use Proposition 4.2 repeatedly. Suppose ak 6= 0 for some k ∈ [4, n − 1].
Then Ψ = 〈αi | ai = 0〉 is contained in Φ(Ak−1Bn−k) and
rΨ > rAk−1Bn−k =
1
2
2nn!
k! 2n−k(n− k)!
2n− 2(n− k)
2n
= 2k−1
(
n
k
)
k
n
.
We have rA3Bn−4 =
4
3
(n − 1)(n − 2)(n − 3) > n(2n + 1) = dimG for n > 6, hence (†)
holds. Since rAk−1Bn−k > rA3Bn−4 for k ∈ [4, n− 1], we see that condition (i) ensures that
(†) holds.
Similarly for condition (ii). If an 6= 0 then rΨ > rAn−1 = 2
n−1 > dimG for n > 9.
Suppose that both a1 6= 0 and an 6= 0. Then rΨ > rAn−2 = 2
n−1n which exceeds dimG
for n > 5. It is clear that rA2An−4 > rA1An−3 > rAn−2 for n > 5 so we may not have both
ai 6= 0 for i ∈ [1, 3] and an 6= 0. Thus condition (iii) ensures that (†) holds.
101
For condition (iv) we need to calculate rA2B1 = 128 and rA21B1 = 192 for n = 5; these
both exceed dimB5(K) = 55.
Suppose that both a2 6= 0 and a3 6= 0. Then (†) holds for n > 5 since then we have
rA1Bn−3 = 6(n− 1)(n− 2) > dimG. The same holds if both a1 6= 0 and a3 6= 0.
The modules of dimension at most dimG + 2 are L(ω1) and L(ωn) for n ∈ [2, 6], the
adjoint modules L(ω2) for n > 3 and L(2ω2) for n = 2 6= p, and L(ω1 + ω2) for n = 2
with p = 5. The only modules with dimensions strictly between dimB = n(n + 1) and
dimG+ 2 are L(ωn) for n ∈ [5, 6] and L(ω1 + ω2) for n = 2 with p = 5.
Lemma 6.3. Suppose that dimV > dimG + 2. If n > 2 then (†) holds except possibly
for the modules L(2ω1) (p > 2), L(ω1 + ω2) and L(3ω1) (p > 3) for n ∈ [2,∞), L(ω3) for
n ∈ [4,∞), L(ωn) for n ∈ [7, 8], L(2ωn) (p > 2) for n ∈ [3, 4], L(ω1 + ω3) for n = 3, and
L(ω1 + 2ω2) (p > 2), L(3ω2) (p > 3) and L(4ω2) (p > 3) for n = 2.
Proof. We shall split the following analysis of weights into five parts according as n > 6
or n = 5, 4, 3 or 2.
Case I: n > 6. By conditions (i), (ii) and (iii) of Proposition 6.2 we may assume that
µ =
∑3
i=1 aiωi if n > 6 or µ = anωn if n ∈ [6, 8] since otherwise (†) holds. Moreover, in the
latter case, we may assume that an = 1 because 2ωn > ωn−1 and (†) holds for L(ωn−1).
The spin module L(ω6) for n = 6 has dimension 64; this is less than dimB6(K) = 78 so
we need only to consider further the spin modules L(ωn) for n ∈ [7, 8]. So we may assume
that µ =
∑3
i=1 aiωi for n > 6. Set m =
∑3
i=1 iai. We follow a similar argument to Lemma
5.3. If m > 4 then we claim that we can find bi with 0 6 bi 6 ai for each i ∈ [1, 3] such
that
µ′ = µ−
(
3∑
i=1
biωi − ωj
)
< µ 6 λ,
where j =
∑3
i=1 ibi and 4 6 j 6 6. Now if m > 4 we need only consider the following
seven weights: 4ω1, 2ω1 + ω2, ω1 + ω3, 2ω2 (m = 4); ω1 + 2ω2, ω2 + ω3 (m = 5); and 2ω3
(m = 6). This is because all other weights have coefficients that are at least that of one of
these seven weights, hence if we can find bi for each of the seven, we can do the same for
102
any other. In fact, if n > 6, since the seven weights µ with m ∈ [4, 6] each satisfy µ > ωm
we can take ai = bi and j = m, whence the claim follows. If n = 6, the claim follows as
before but if m = 6 we must use the partial ordering 2ω3 > 2ω6 > ω5.
Let us take for example the weight 2ω1 + 3ω3. If n > 6 then, since 2ω3 > ω6, we have
2ω1 + 3ω3 > 2ω1 + ω3 + ω6; here j = 6, b1 = a1 = 2, b2 = a2 = 0 and b3 = 1 < 3 = a3. If
n = 6 then 2ω1 + 3ω3 > 2ω1 + ω3 + ω5; the only difference here is that we take j = 5.
We may assume that m 6 3 since otherwise the claim above shows that we can find a
weight µ′ < µ such that a′j 6= 0 and j ∈ [4, 6] for n > 6 or j ∈ [4, 5] for n = 6; hence, by
Premet’s theorem and condition (i) of Proposition 6.2, (†) holds. Thus for n > 6 we need
to consider further the irreducible modules with highest weights 2ω1, ω3, ω1+ω2 and 3ω1.
Case II: n = 5. If a4 6= 0 then (†) holds since ω4 > ω3 and, applying Premet’s theorem,
we have |Λω4| > rA3B1 + rA2B2 = 32 + 24 = 56. Similarly, if a5 6= 0 then (†) holds since
2ω5 > ω4 and the dimension of the spin module L(ω5) is smaller than dimG. Thus we
may assume that µ =
∑3
i=1 aiωi. Suppose that at least two ai for i ∈ [1, 3] are non-zero.
By condition (v) of Proposition 6.2 we may assume that µ = a1ω1 + a2ω2 since (†) holds
otherwise. Since 2ω1 + ω2 > ω1 + ω3 and ω1 + 2ω2 > ω2 + ω3 and (†) holds for both
ω1 + ω3 and ω2 + ω3 (as we previously noted), we need only investigate further the case
a1 = a2 = 1.
Assume that only one ai is non-zero for i ∈ [1, 3]. If a1 6= 0 we may assume that a1 6 3
since 4ω1 > 2ω1 + ω2. The condition (†) holds if a2 6= 0 since 2ω2 > ω1 + ω3 and L(ω2) is
the adjoint module. Similarly if a3 6= 0 we may assume that a3 = 1 since 2ω3 > ω2 + ω3.
Thus we must consider further the irreducible modules with highest weights ω1+ω2, 2ω1,
3ω1 and ω3.
Case III: n = 4. Assume that at least three coefficients of µ are non-zero. Since the
smallest rΨ in this case occurs for Ψ = B1 and rB1 = 72, we see that (†) holds. Suppose
that precisely two ai for i ∈ [1, 4] are non-zero. If both a1 6= 0 and a3 6= 0 then (†) holds
since ω1 + ω3 > ω1 + ω2 and rA1B1 + rB2 = 36 + 12 = 48. The same holds true if both
a1 6= 0 and a4 6= 0 since ω1 + ω4 > ω4 and rA2 + rA3 = 32 + 8 = 40, and hence if both
a2 6= 0 and a3 6= 0 since ω2 + ω3 > ω1 + 2ω4. Since rA2
1
= 48 we cannot have both a2 6= 0
103
and a4 6= 0 and hence we cannot have both a3 6= 0 and a4 6= 0 since ω3 + ω4 > ω2 + ω4.
Finally, if both a1 6= 0 and a2 6= 0 then (†) holds if a1 + a2 > 2 since 2ω1 + ω2 > ω1 + ω3
and ω1 + 2ω2 > ω2 + ω3.
Assume now that only one coefficient of µ is non-zero. If a1 6= 0 then we may assume
that a1 6 3 since 4ω1 > 2ω1 + ω2. If a2 6= 0 then (†) holds since 2ω2 > ω1 + ω3 and
L(ω2) is the adjoint module which we do not consider by assumption. If a3 6= 0 we may
assume that a3 = 1 since 2ω3 > ω1 + ω3. Similarly, if a4 6= 0 we may assume that a4 = 2
since 3ω4 > ω3 + ω4 and the spin module L(ω4) only has dimension 16. Thus further
investigation is required for the irreducible modules with highest weights 2ω1, ω3, ω1+ω2,
3ω1 and 2ω4.
Case IV: n = 3. Since r∅ = 24 we can assume that at most two ai for i ∈ [1, 3] are non-
zero. By performing quick calculations we have rA1 = 12, rB1 = 8 and rA1B1 = rA2 = 4.
Consider the case that precisely two coefficients of µ are non-zero. We see that (†) holds
if both a2 and a3 are non-zero since ω2 + ω3 > ω1 + ω3. If a1 and a2 are non-zero
then the partial ordering ω1 + 2ω2 > 2ω1 + ω2 > 2ω2 > ω1 + 2ω3 shows that (†) holds
unless a1 = a2 = 1. Similarly, if a1 and a3 are non-zero then the partial orderings
ω1 + 2ω3 > ω1 + ω2 > 2ω3 and 2ω1 + ω3 > ω2 + ω3 together show that (†) holds unless
a1 = a3 = 1. Now assume that only one coefficient of µ is non-zero. If a1 6= 0 then we
may assume that a1 6 3 since 4ω1 > 2ω1 + ω2 and we saw previously that (†) holds for
2ω1+ω2. Similarly, if a2 6= 0 then (†) holds since 2ω2 > ω1+2ω3 and L(ω2) is the adjoint
module, and if a3 6= 0 then we may assume that a3 = 2 since 3ω3 > ω2 + ω3 and the spin
module L(ω3) has dimension less than G. Thus we need to consider further the weights
ω1 + ω2, ω1 + ω3, 2ω1, 3ω1 and 2ω3.
Case V: n = 2. We calculate r∅ = 4, rA1 = 2 and rB1 = 1. Assume that both a1 and
a2 are non-zero. Since 2ω1 + ω2 > 3ω2 > ω1 + ω2 > ω2 and ω1 + 3ω2 > 2ω1 + ω2 > 3ω2 >
ω1+ ω2 > ω2, we need only consider further the weights with a1 = 1 and a2 6 2. Assume
that a1 6= 0 and a2 = 0. We may take a1 6 3 since 4ω1 > 2ω1 + 2ω2; this follows from
the partial ordering for 2ω1+ω2 just given together with Premet’s theorem. Assume now
that a1 = 0 and a2 6= 0. We may take a2 6 4 since 5ω2 > ω1 + 3ω2. Thus we need to
104
consider further the weights ω1 + ω2, ω1 + 2ω2, 2ω1, 3ω1, 3ω2 and 4ω2 for n = 2.
6.2 Even characteristic
In this section we deal with the case p = 2.
Lemma 6.4. Let G = Spin2n+1(K) act on V = L(λ) for p = 2 and dimV > dimG + 2.
Then (†) holds with the possible exceptions of L(ω3) for n ∈ [4,∞), L(ω1+ω2) for n > 2,
L(ωn) for n ∈ [7, 8], L(ω4) for n = 5 and L(ω1 + ω3) for n = 3.
Proof. We write the 2-restricted highest weight λ =
∑n
i=1 aiωi with ai ∈ {0, 1} for all
i ∈ [1, n]. We shall treat ωn differently from the other fundamental weights since ωi =
α1+2α2+ · · ·+(i−1)αi−1+i (αi + · · ·+ αn) for i < n and ωn =
1
2
(α1 + 2α2 + · · ·+ nαn).
We shall assume in turn that the number of non-zero ai is one, then two, and then three
or more.
Suppose that only one of the ai is non-zero; we may assume that i > 3 as otherwise
the module has dimension at most dimG. We have |W.ωi| = 2
i
(
n
i
)
.
The weights of the module L(ωn) are precisely the 2
n weights in W.ωn. Since this is
true for all p we may refer back to the consideration of this module in the previous section
where we see that we need to further examine L(ωn) for n ∈ [7, 8].
Henceforth we may assume that i < n. We see that 〈ωi, α〉 = 0 or ±2 for short
roots α ∈ Φ. Moreover, 〈ωi, α〉 = 2 for i such α. Therefore, for a fixed short root
α there are σi weights µ ∈ W.ωi with weight string containing the weights µ µ + 2α
(but possibly not µ + α), where σi = 2
i−1
(
n−1
i−1
)
. Now if 4 6 i 6 n − 1 then, since
σi > σ4 = 2
3
(
n−1
3
)
> n(2n+ 1) = dimG for n > 6, it remains for us to consider L(ω4) for
n = 5 as well as L(ω3) for n > 4.
Next suppose that precisely two coefficients ai of λ are non-zero. Consider the module
L(ωi + ωj) where
|W.(ωi + ωj)| =
2jn!
i!(j − i)!(n− j)!
.
Assume that 1 6 i < j 6 n− 1 so that 〈ωi + ωj, α〉 = 0,±2 or ±4 for short roots α ∈ Φ.
105
Moreover, 〈ωi + ωj, α〉 = 2 for j − i short roots and 〈ωi + ωj, α〉 = 4 for i short roots.
Thus, for a given α there are τij weights µ ∈ W.(ωi + ωj) with weight string containing
the weights µ µ+ 2α or µ µ+ 4α where
τij =
|W.(ωi + ωj)|(j − i+ i)
2n
=
2j−1j(n− 1) · · · (n− j + 1)
i!(j − i)!
.
As usual we take α /∈ Φs and subsequently apply the adjacency principle. We may do
this since p = 2 whence 2kα(s) = 1 if and only if α(s) = 1.
Assume that (i, j) 6= (1, 2). It is clear that τij > τ13 = τ23 = 6(n − 1)(n − 2). Since
this exceeds dimG for n > 5, the modules L(ω1+ω3) and L(ω2+ω3) for n = 4 as well as
L(ω1 + ω2) for n > 3 require further consideration.
Now take j = n in which case 〈ωi+ωn, α〉 = ±1 or ±3. We concentrate on the former
case since then we may apply the adjacency principle. There are n− i short roots α such
that 〈ωi+ωn, α〉 = 1. Thus, for a given short root α there are θi weights µ ∈W.(ωi+ωn)
with weight string containing µ µ + α where θi = 2
n−1
(
n−1
i
)
. Now θi > θ1 = 2
n−1(n− 1)
for i ∈ [1, n − 2] which exceeds dimG for n > 5. If i = n − 1 we conclude for n > 5 by
applying Proposition 4.3. Therefore it remains to consider ω1 + ωn for n ∈ [2, 4]. Both
θ2 = 2
n−2(n− 1)(n− 2) and θ3 =
2n−2
3
(n− 1)(n− 2)(n− 3) exceed dimG for n > 5 so we
must later deal with L(ω2 + ωn) for n ∈ [3, 4] and L(ω3 + ω4) for n = 4.
We can deal with the modules L(ωi+ωj) for n = 4 with (i, j) 6= (1, 2) and L(ω2+ω3)
for n = 3 by referring to Lu¨beck’s tables [18]. First assume that n = 4. Consider
L(ω1 + ω4) where we see that there are no dominant weights below ω1 + ω4 in the partial
ordering appearing with multiplicity zero, i.e., the conclusion of Premet’s theorem holds.
Since ω1 + ω4 > ω4 and we calculate rA2 = 32 and rA3 = 8 we conclude that (†) holds for
this module. Indeed, by the same reasoning the same conclusion holds for the modules
L(ω3 + ω4) and L(ω2 + ω4); we only need to calculate that rA2
1
= 48. We see from
the tables that the conclusion of Premet’s theorem holds for neither L(ω1 + ω3) nor
L(ω2 + ω3). We see that ω1 + ω3 > 2ω4 and m2ω4 6= 0 for L(ω1 + ω3) from the tables and
we calculate |W.(ω1 + ω3)| = 96 and |W.(2ω4)| = 16. There are three short roots α for
106
which 〈ω1+ω3, α〉 = 2 or 4 and we have 〈2ω4, α〉 = 2 for all the positive short roots. Thus,
for a given α, there are 96.3
8
= 36 weights µ ∈ W.(ω1 + ω3) with weight string containing
µ µ+2α or µ µ+4α. Similarly there are 16.4
8
= 8 weights ν ∈W.(2ω4) with weight string
containing ν ν+2α. Thus |Λ| > 44 whence (†) holds. In an exactly analogous way we can
handle L(ω2 + ω3) by using the Weyl group orbit of ω1 + 2ω4 which occurs with non-zero
multiplicity in this module.
Assume now that n = 3 and consider the module L(ω2+ω3). From Lu¨beck’s tables we
find that the conclusion of Premet’s theorem holds for this module. Since ω2+ω3 > ω1+ω3
and rA1 = 12 we have |Λ| > 24 > dimG, so (†) holds.
Suppose that three coefficients of λ are non-zero and that n > 4. Assume first that
an = 0. Then 〈λ, α〉 = ±2,±4 or ±6 for short roots α and, as before, we are only
interested in the first two of these possibilities. For a fixed α the minimum number
of weights µ ∈ W.λ such that 〈µ, α〉 = 2 or 4 occurs for λ = ω1 + ω2 + ω3; this is
23(n− 1)(n− 2) which exceeds dimG for n > 4. We notice that the Weyl group orbit of
a dominant weight increases in size as more of its coefficients (when written as a Z-linear
combination of fundamental weights) are non-zero. It follows, therefore, that (†) holds
for all dominant weights λ ∈ X(T ) with three or more coefficients ai non-zero for i 6= n.
Assume instead that an = 1, so λ = ωi + ωj + ωn where 1 6 i < j 6 n − 1. Then
〈λ, α〉 = ±1,±3 or ±5 for short roots α and we can only apply the adjacency principle
in the first of these possibilities. The minimum number of µ ∈ W.λ with weight string
µ µ + α occurs for either λ = ω1 + ωn−1 + ωn or ωn−2 + ωn−1 + ωn; this is 2
n−1(n − 1)
which exceeds dimG for n > 5. Therefore (†) holds for all weights λ with three or more
coefficients ai non-zero including an 6= 0 provided n > 5.
It remains to consider the irreducible modules L(ω1 + ω2 + ω3 + ω4), L(ω1 + ω2 + ω4),
L(ω1 + ω3 + ω4) and L(ω2 + ω3 + ω4) all for n = 4 and L(ω1 + ω2 + ω3) for n = 3.
Consider L(ω1 + ω2 + ω3 + ω4) for n = 4. We have 〈ω1 + ω2 + ω3 + ω4, α〉 = 1 for
one short root only, namely α4. For α = α4 there are
244!
2.4
= 48 weights µ ∈ W.
(∑4
i=1 ωi
)
with weight string µ µ+ α. Thus (†) holds for this module since |Λ| > 48 > 36 = dimG.
Recall that, by Theorem 2.4, the remaining four modules are tensor decomposable and
107
are of the form L(ν)⊗L(ωn) where ν ∈ X(T )L. For L(ωn) we find that 〈ωn, α〉 = 1 for all
positive short roots α. Thus, if we fix a short root α and take this to lie outside Φs then
all weights occur in pairs differing by α, i.e., we have |Λωn| > 2
n−1. (We shall study this
module further in Lemma 6.7.) We appeal to Proposition 4.3 for the remaining modules.
From Lu¨beck’s tables [18] we find that dimL(ω1 + ω3) = 246 and dimL(ω2 + ω3) = 784
for n = 4, and dimL(ω1+ω2) = 64 or 160 according as n = 3 or 4. Thus, for each module
we have |Λωn| dimL(ν) > dimG, whence (†) holds.
6.3 Weight string analysis
We list in Table 6.1 the weights which require further consideration following Lemmas 6.3
and 6.4.
λ n Lemma
2ω1 [2,∞) (p 6= 2) -
ω3 [4,∞) 6.5 (♦)
ω1 + ω2 [6,∞) 6.5 (♦)
[3, 5] 6.5 (†) p 6= 3, (♦) p = 3
2 (p 6= 5) 6.5 (♦)
3ω1 [6,∞) (p > 3) 6.5 (♦)
[2, 5] (p > 3) 6.5 (♦)
ωn [7, 8] 6.7 (♦)
ω4 5 (p = 2) 6.6 (♦)
2ωn 4 (p 6= 2) 6.9 (†)
3 (p 6= 2) 6.9 (♦)
ω1 + ω3 3 6.8 (†) p 6= 7, (♦) p = 7
ω1 + 2ω2 2 (p 6= 2) 6.8 (†) p 6= 2, 3, (♦) p = 3
3ω2 2 (p > 3) 6.8 (♦)
4ω2 2 (p > 3) 6.8 (†)
Table 6.1: Possible weights in type Bn for n > 2
We begin by considering the modules L(ω3) for n > 4 and L(ω1 + ω2) and L(3ω1) for
n > 2.
Recall from [2, p.47] that for the root system of type Bn we have Φ = ΦS ∪ΦL where
ΦL = {±ǫi±ǫj | 1 6 i < j 6 n} consists of 2n(n−1) long roots and ΦS = {±ǫi | 1 6 i 6 n}
108
consists of 2n short roots. Let αi = ǫi − ǫi+1 for i ∈ [1, n− 1] and αn = ǫn be the simple
roots. Therefore we have ωi =
∑i
k=1 ǫk for i < n and ωn =
1
2
∑n
k=1 ǫk. We shall need this
for the next lemma.
Lemma 6.5. Let G act on the irreducible module V where we take V to be one of L(ω3)
for n > 4, L(ω1 + ω2) for n > 2, or L(3ω1) for n > 2 with p > 3. Then (♦) is satisfied
for all X ∈ I unless V = L(ω1 + ω2) for n ∈ [3, 5] with p 6= 3 in which case (†) holds or
for n = 2 with p = 5 in which case the module is small.
Proof. If p = 2 then the conclusion of Premet’s theorem does not hold. For L(ω3) we
see from Lu¨beck’s tables that when n 6 11 we have mω2 = 0. The tables also show that
mω1 6= 0; in order to conclude that this is true for n > 11 we can adapt the arguments
in Section 2.4 of [29]. Zalesski’s technique involves rewriting weights in Cn as weights in
Dn and shows in Lemma 12 of his paper that for a dominant weight µ =
∑n
i=1 aiωi in Cn,
provided an = 0, two dominant weights λ < µ in Cn remain dominant when considered in
Dn and the partial ordering is preserved. Then, since the conclusion of Premet’s theorem
holds for groups of type Dn in characteristic 2, he concludes that λ appears with non-zero
multiplicity in Π(µ) in Cn. We rewrite the weights ω1 and ω3 in Bn as weights in Dn and
apply the theory in Section 2.4 in order to conclude that each weight in the Weyl group
orbit of ω1 appears with non-zero multiplicity in Π(ω3) in Bn. Analogously, for L(ω1+ω2)
with p = 2 we can conclude that both mω3 and mω1 are non-zero.
Consider the module L(ω3) for n > 4. The usual calculations are detailed in Figure
6.1.
Assume that p 6= 2. We see from Figure 6.1 that for n > 4 we have |Λ| > 2(n − 1)2.
We may assume that there are two orthogonal roots (α short and β long) not in Φs since
(♦) holds for both X = Bn−1 (e
Bn−1
ω3
= 4n− 2) and Dn (e
Dn
ω3
= 2n). It is straightforward
to check that 〈ω3, α〉 = 0 or ±2 and 〈ω3, β〉 = 0, ±1 or ±2. We note that for a root
system of type Bn there are 2n.2(n− 1)(n− 2) orthogonal pairs of roots α short, β long.
We have 〈ω3, α〉 = 2 and 〈ω3, β〉 = 2 for 3 pairs of roots α, β with (α, β) = 0.
In more detail, the 3 pairs of orthogonal long and short roots are ǫi and ǫj + ǫk with
109
i ω |W.ω| mω
n 6 11, p 6= 2 n > 12, p 6= 2 p = 2
3 ω3
4
3
n(n− 1)(n− 2) 1 1 1
2 ω2 2n(n− 1) 1 1 0
1 ω1 2n n− 1 1 > 1
0 0 1 n 1 0
Weight No. of l
strings strings n 6 11, p 6= 2 n > 12, p 6= 2 p = 2
µ3
4
3
(n− 1)(n− 2)(n− 3)
µ3 µ2 µ3 2(n− 1)(n− 2) 2(n− 1)(n− 2) 2(n− 1)(n− 2) 2(n− 1)(n− 2)
µ2 µ1 µ2 2(n− 1) 4(n− 1) 2(n− 1)
µ1 µ0 µ1 1 n 1 1
Lower bound on |Λ| 2n2 − n 2n2 − 4n+ 3 2n2 − 6n+ 5
Figure 6.1: λ = ω3 for n > 4
i, j, k ∈ [1, 3], j < k and i 6= j 6= k 6= i. There are 4n(n− 1)(n− 2) triples (µ;α, β) with
〈µ, α〉 = 〈µ, β〉 = 2 for orthogonal roots α ∈ ΦS, β ∈ ΦL and µ ∈ W.ω3. For a given pair
α, β there is 1 weight µ ∈ W.ω3 such that 〈µ, α〉 = 〈µ, β〉 = 2. Thus there is a 3 × 3
weight net.
We have 〈ω3, α〉 = 0 and 〈ω3, β〉 = 2 for 6(n − 3) pairs of roots α, β with (α, β) = 0.
These pairs of orthogonal long and short roots are ǫi + ǫj with 1 6 i < j 6 3 and ±ǫk
with k ∈ [4, n]. There are 8n(n − 1)(n − 2)(n − 3) triples (µ;α, β) with 〈µ, α〉 = 0 and
〈µ, β〉 = 2 for orthogonal roots α ∈ ΦS, β ∈ ΦL and µ ∈ W.ω3. For a given pair α, β
there are 2(n− 3) weights µ ∈ W.ω3 such that 〈µ, α〉 = 0 and 〈µ, β〉 = 2. Thus there are
2(n− 3) weight nets of size 1× 3; it is easy to check that these take the form µ3 µ1 µ3.
Similarly we calculate that there are 4(n− 3) weight nets of size 3× 2, 4(n− 3)(n− 4)
weight nets of size 1× 2 and 2+ 2(n− 3)(n− 4) weight nets of size 3× 1 (which take the
form µ3 µ2 µ3 as indicated by Figure 6.1). We need not consider weights µ ∈ W.ω3 with
〈µ, α〉 = 〈µ, β〉 = 0 since there is no contribution to |Λ| from 1× 1 weight nets. So from
weight nets containing µ ∈ W.ω3 we find that
|Λ| > 4 + 2(n− 3) + 4(n− 3) + 4(n− 3)(n− 4) + 2 + 2(n− 3)(n− 4) = 6n2 − 28n+ 36.
Now we consider the remaining weight nets for L(ω3), i.e., those containing no weights
from W.ω3. As before, we can check that 〈ω2, α〉 = 0 or ±2 and 〈ω2, β〉 = 0, ±1 or ±2. It
is not possible to find pairs of orthogonal roots α and β with 〈ω2, α〉 = 〈ω2, β〉 = 2. We
110
find that there are two 3× 2 and 2(n− 3) weight nets of size 3× 1. All other weight nets
and strings are included in those found previously.
In summary, the weight nets are provided below.
µ3 µ2 µ3 µ3 µ3 µ2 µ3 µ3 µ3 µ2 µ3 µ2 µ1 µ2
µ1 µ0 µ1 µ2 µ1 µ2 µ3 µ2 µ3 µ3 µ2 µ1 µ2
µ3 µ2 µ3 µ3
1 2(n− 3) 4(n− 3) 4(n− 3)(n− 4) 2(n2 − 7n+ 13) 2
Since all weights have multiplicity at least one we find |Λ| > 6n2 − 28n+ 42 > 2n2 =
|Φ(G)| for n > 5. We are therefore left to consider L(ω3) for n = 4; here we know that
mω1 = 3 and m0 = 4, so using the weight nets above we have |Λ| > 38 > 32 = |Φ(B4)|.
Thus (♦) is satisfied for all X ∈ I provided p 6= 2.
Assume now that p = 2. From Figure 6.1 we see that |Λ| > 2n2 − 6n+ 5. If X = Dn
then (♦) is satisfied for n > 4. If X = Bn−1 then (♦) is satisfied provided n > 5. For
n = 4 when X = B3 there are three clusters. They are 1 · · ·, 0 · · · and −1 · · · of sizes
14, 20 and 14 respectively. Since eB3ω3 = 14 we see the (♦) is satisfied for B3. Therefore
we can assume that there are two orthogonal roots, one long and one short, not in Φs.
Using the weight nets described above for p 6= 2 and the fact that α(s) = 1 if and only if
2α(s) = 1 we see that |Λ| > 6n2 − 32n+ 49. This exceeds |Φ(G)| for n > 6.
We are left to consider L(ω3) for n ∈ [4, 5] with p = 2. From Lu¨beck’s tables we
see that mω1 = 2 in both cases. Using the weight nets above we see that if n = 5 then
|Λ| > 46 so that (♦) is satisfied provided |Φ(G)| > 4, and if n = 4 then |Λ| > 22 so that
(♦) is satisfied provided |Φ(G)| > 10. Thus, if n = 5 we are left to consider X = ∅, A1,
B1, A
2
1, A1B1 and D2 and if n = 4 we are left with X = ∅, A1, A
2
1, D2, D2A1, A2, B1,
B1A1, B1D2, B1A2, B2 and B2A1. We shall provide the clusters and cliques for the six
remaining cases when n = 5; we omit the calculations for n = 4.
Let X = A21 with simple roots α1 and α3. By using the cliques tabulated below
together with the negative counterparts of the cliques in the second column we have
d
A2
1
ω3 > 48 > 46 = e
A2
1
ω3 .
111
(λ, n, X) = (ω3, 5, A21)
Cliques Cluster size l Cliques Cluster size l
· 1 1 1 1 4 16 · 1 · 2 1 4 4
0 0 · 1 1 4 · 1 · 1 1 4
0 0 0 0 1 2 · 1 · 0 1 4 4
0 0 0 0 -1 2 · 1 · 0 -1 4
0 0 · -1 -1 4 1 2 · 3 3 2 2
· -1 -1 -1 -1 4 · 1 2 3 3 2
· 0 · 1 1 4 4 1 2 2 2 3 1 3
· 0 · -1 -1 4 1 2 2 2 1 1
· 0 0 0 1 2 2 1 2 2 1 1 1
· 0 0 0 -1 2 1 2 1 1 1 1
Let X = A1B1 with simple roots α1 and α5. By using the cliques tabulated below
together with the negative counterparts of the cliques in the second column we have
dA1B1ω3 > 48 > 46 = e
A1B1
ω3
.
(λ, n, X) = (ω3, 5, A1B1)
Cliques Cluster size l Cliques Cluster size l
· 1 1 1 1 2 16 · 1 1 0 · 4 4
0 0 1 1 1 1 · 1 0 0 · 4
0 0 0 1 1 1 1 2 2 2 · 2 6
0 0 0 0 · 2 · 1 2 2 · 4
0 0 0 -1 -1 1 · 1 1 2 · 4
0 0 -1 -1 -1 1 1 2 3 3 3 1 2
· -1 -1 -1 -1 2 1 2 2 3 3 1
· 0 1 1 1 2 8 · 1 2 3 3 2
· 0 0 1 1 2
· 0 0 0 · 4
· 0 0 -1 -1 2
· 0 -1 -1 -1 2
LetX = D2 with simple root α1 and the longest root α0. By using the cliques tabulated
below together with the negative counterparts of the cliques in the second column we have
dD2ω3 > 48 > 46 = e
D2
ω3
. Note that we provide a representative weight from each cluster.
112
(λ, n, X) = (ω3, 5, D2)
Cliques Cluster size l Cliques Cluster size l
1 1 1 1 1 8 24 1 0 1 1 1 2 4
0 0 1 1 1 4 1 0 0 1 1 2
0 0 0 1 1 4 1 0 0 0 1 2
0 0 0 0 1 4 1 1 1 0 1 4 4
0 0 0 0 -1 4 1 1 0 0 1 4
0 0 0 -1 -1 4 1 1 1 0 -1 4 4
0 0 -1 -1 -1 4 1 1 0 0 -1 4
Let X = B1 with simple root α5. By using the cliques tabulated below together
with the negative counterparts of the cliques in the second and third columns we have
dB1ω3 > 50 > 48 = e
B1
ω3
.
(λ, n, X) = (ω3, 5, B1)
Cliques Cluster size l Cliques Cluster size l Cliques Cluster size l
1 1 1 1 1 2 16 1 1 1 0 · 2 4 1 2 3 3 3 1 3
0 1 1 1 1 2 1 1 0 0 · 2 1 2 2 3 3 1
0 0 1 1 1 2 1 0 0 0 · 2 1 1 2 3 3 1
0 0 0 1 1 2 1 1 1 2 · 2 4 0 1 2 3 3 1
0 0 0 0 · 4 0 1 1 2 · 2 1 2 2 1 1 1 2
0 0 0 -1 -1 2 0 0 1 2 · 2 1 1 2 1 1 1
0 0 -1 -1 -1 2 1 2 2 2 · 2 4 0 1 2 1 1 1
0 -1 -1 -1 -1 2 1 1 2 2 · 2
-1 -1 -1 -1 -1 2 0 1 2 2 · 2
Let X = A1 with simple root α1. By using the cliques tabulated below together
with the negative counterparts of the cliques in the first and second columns we have
dA1ω3 > 50 > 48 = e
A1
ω3
.
(λ, n, X) = (ω3, 5, A1)
Cliques Cluster size l Cliques Cluster size l Cliques Cluster size l
1 2 3 3 3 1 2 · 0 1 1 1 2 4 · 1 1 1 1 4 16
1 2 2 3 3 1 · 0 0 1 1 2 · 0 1 1 1 2
· 1 2 3 3 2 · 0 0 0 1 2 · 0 0 1 1 2
1 2 2 2 3 1 3 · 1 1 0 1 2 2 · 0 0 0 1 2
· 1 2 3 3 2 · 1 0 0 1 2 · 0 0 0 -1 2
· 1 1 2 3 2 · 1 1 0 -1 2 2 · 0 0 -1 -1 2
· 1 1 2 1 2 2 · 1 0 0 -1 2 · 0 -1 -1 -1 2
· 1 0 1 1 2 · -1 -1 -1 -1 4
· 1 2 2 1 2 2
· 1 2 1 1 2
113
Let X = ∅; by using the cliques tabulated below together with their negative coun-
terparts we have d∅ω3 > 52 > 50 = e
∅
ω3
.
(λ, n, X) = (ω3, 5, ∅)
Cliques Cluster size l Cliques Cluster size l Cliques Cluster size l
1 1 1 1 1 2 8 1 2 2 2 1 1 2 1 2 3 3 3 1 3
0 1 1 1 1 2 1 2 2 1 1 1 1 2 2 3 3 1
0 0 1 1 1 2 1 2 1 1 1 1 1 1 2 3 3 1
0 0 0 1 1 2 1 0 1 1 1 1 2 0 1 2 3 3 1
0 0 0 0 1 2 1 0 0 1 1 1 1 1 1 2 1 1 2
1 2 2 2 3 1 2 1 0 0 0 1 1 0 1 1 2 1 1
1 1 2 2 3 1 1 1 1 2 3 1 2 0 0 1 2 1 1
0 1 2 2 3 1 0 1 1 2 3 1 1 1 1 0 -1 1 2
1 1 1 0 1 1 2 0 0 1 2 3 1 0 1 1 0 -1 1
0 1 1 0 1 1 1 1 2 2 1 1 1 0 0 1 0 -1 1
0 0 1 0 1 1 0 1 2 2 1 1
Suppose that n > 6. Using the work for L(ω3) above together with Premet’s theorem,
we can conclude for L(ω1 + ω2) for all p and L(3ω1) for p 6= 2 that (♦) is satisfied for all
X ∈ I.
Consider L(ω1+ω2) for n 6 5. If n ∈ [4, 5] then Figure 6.2 shows that (†) holds unless
p = 3 when (♦) is satisfied for each X ∈ I.
i ω |W.ω| mω
p 6= 2, 3, p 6= 5 = n p = 3 p = 5 = n p = 2
5 ω1 + ω2 4n(n− 1) 1 1 1 1
4 ω3
4
3
n(n− 1)(n− 2) 2 1 2 2
3 2ω1 2n 1 1 1 0
2 ω2 2n(n− 1) 2 1 2 0
1 ω1 2n 2n− 1 n 8 2n− 2
0 0 1 2n− 1 n− 1 8 0
Weight No. of l
strings strings p 6= 2, 3, p 6= 5 = n p = 3 p = 5 = n p = 2
µ5 4(n− 1)(n− 2)
µ5 µ3 µ5 2(n− 1) 2(n− 1) 2(n− 1) 2(n− 1) 2(n− 1)
µ5 µ2 µ1 µ2 µ5 2(n− 1) 8(n− 1) 4(n− 1) 8(n− 1) 4(n− 1)
µ4
4
3
(n− 1)(n− 2)(n− 3)
µ4 µ2 µ4 2(n− 1)(n− 2) 4(n− 1)(n− 2) 2(n− 1)(n− 2) 4(n− 1)(n− 2) 4(n− 1)(n− 2)
µ3 µ1 µ0 µ1 µ3 1 2n+ 1 n+ 1 2n 2n− 2
Lower bound on |Λ| 4n2 − 1 2n2 + n− 1 4n2 − 2 4n(n− 1)
Figure 6.2: λ = ω1 + ω2 for n ∈ [4, 5]
If n = 3 then Figure 6.3 shows that (†) holds unless p = 3 in which case (♦) is satisfied
for each X ∈ I.
114
i ω |W.ω| mω
p 6= 2, 3 p = 3 p = 2
5 ω1 + ω2 24 1 1 1
4 2ω3 8 2 1 2
3 2ω1 6 1 1 0
2 ω2 12 2 1 0
1 ω1 6 5 2 4
0 0 1 5 1 0
Weight No. of l
strings strings p 6= 2, 3 p = 3 p = 2
µ5 8
µ5 µ3 µ5 4 4 4 4
µ5 µ2 µ1 µ2 µ5 4 16 8 8
µ4 µ2 µ4 4 8 4 8
µ3 µ1 µ0 µ1 µ3 1 7 3 4
Lower bound on |Λ| 35 19 24
Figure 6.3: (λ, n) = (ω1 + ω2, 3)
If n = 2 then Figure 6.4 shows that (♦) is satisfied for each X ∈ I other than X = ∅
for p 6= 2, 5, and A1 and B1 for p = 5.
i ω |W.ω| mω
p 6= 5 p = 5
2 ω1 + ω2 8 1 1
1 ω2 4 2 1
Weight No. of l
strings strings p 6= 2, 5 p = 5 p = 2
µ2 µ2 2 2 2 2
µ2 µ1 µ1 µ2 2 6 4 8
Lower bound on |Λ| 8 6 10
Figure 6.4: (λ, n) = (ω1 + ω2, 2)
If X = ∅ and p 6= 5 then the following cliques show that d∅ω1+ω2 > 9 in which case (♦)
holds. Note that the weights in W.ω2 have multiplicity 2 and are
1
2
1, 1
2
0, -1
2
0 and -1
2
-1.
1
2
1 1
2
-1 3
2
2 1
2
2 - 3
2
-1
1
2
0 - 1
2
-1 3
2
1 - 1
2
1 - 3
2
-2
- 1
2
0 - 1
2
-2
If p = 5 then the module is small since it has dimension dimG+2. Indeed, we can see
forX ∈ {∅, A1, B1} there are possible configurations of weights such that dXω1+ω2 = e
X
ω1+ω2
.
Now consider L(3ω1) for n ∈ [3, 5]. From Figure 6.5 below we see that we may take
two orthogonal roots (α short and β long) not in Φs. If n = 3 the Weyl group orbit
W.(2ω3) is present in place of W.ω3 and 2n+ 3 is not a prime.
We see that 〈3ω1, α〉 can take the value 0 or ±6 and 〈3ω1, β〉 can take the value 0
or ±3. Similarly we can have 〈ω1 + ω2, α〉 = 0, ±2 or ±4 and 〈ω1 + ω2, β〉 = 0, ±1,
±2 or ±3 though for a fixed pair α, β there are some pairs of these values that are not
possible. It is sufficient to only consider the weight nets formed from weights in W.(3ω1)
and W.(ω1 + ω2); these are detailed below and the number of each such is given.
115
i ω |W.ω| mω
p 6= 2, 3, 2n+ 3 p = 2n+ 3
6 3ω1 2n 1 1
5 ω1 + ω2 4n(n− 1) 1 1
4 ω3 (2ω3)
4
3
n(n− 1)(n− 2) 1 1
3 2ω1 2n 1 1
2 ω2 2n(n− 1) 1 1
1 ω1 2n n n− 1
0 0 1 n n− 1
Weight No. of l
strings strings p 6= 2, 3, 2n+ 3 p = 2n+ 3
µ6 2(n− 1)
µ6 µ3 µ1 µ0 µ1 µ3 µ6 1 n+ 2 n+ 1
µ5 4(n− 1)(n− 2)
µ5 µ3 µ5 2(n− 1) 2(n− 1) 2(n− 1)
µ5 µ2 µ1 µ2 µ5 2(n− 1) 4(n− 1) 4(n− 1)
µ4
4
3
(n− 1)(n− 2)(n− 3)
µ4 µ2 µ4 12 24 24
Lower bound on |Λ| 2n2 + n 2n2 + n− 1
Figure 6.5: λ = 3ω1 for n ∈ [3, 5]
µ6 µ6 µ3 µ1 µ0 µ1 µ3 µ6 µ5 µ5 µ2 µ1 µ2 µ5 µ5 µ2 µ1 µ2 µ5 µ5 µ3 µ5 µ5 µ3 µ5 µ5 µ5
µ5 µ1 µ5 µ2 µ1 µ2 µ5 µ4 µ2 µ4 µ4 µ5
µ6 µ1 µ5 µ3 µ5 µ5
µ5
2 1 2 2 2(n− 3) 2 2(n− 3) 4(n− 3) 4(n− 3)
Therefore |Λ| > 19n− 20 for p 6= 2, 3 or 2n+ 3 and |Λ| > 19n− 25 for p = 2n+ 3. In
either case (♦) is satisfied for all X ∈ I for n ∈ [3, 5].
If n = 2 then we show in Figure 6.6 that (♦) holds for all X ∈ I.
i ω |W.ω| mω
p 6= 2, 3, 7 p = 7
5 3ω1 4 1 1
4 ω1 + 2ω2 8 1 1
3 2ω1 4 1 1
2 2ω2 4 1 1
1 ω1 4 2 1
0 0 1 2 1
Weight No. of l
strings strings p 6= 2, 3, 7 p = 7
µ5 2
µ5 µ3 µ1 µ0 µ1 µ3 µ5 1 4 3
µ4 µ3 µ4 2 2 2
µ4 µ2 µ1 µ2 µ4 2 4 4
Lower bound on |Λ| 10 9
Figure 6.6: (λ, n) = (3ω1, 2)
We note that we are unable to draw conclusions for L(ω1 + ω2) and L(3ω1) with
n ∈ [4, 5] by appealing to Premet’s theorem since the multiplicities of weights in W.ω1
and 0 in L(ω1 + ω2) and L(3ω1) are not always greater than those for the corresponding
weights in L(ω3). Indeed, using the multiplicities given for W.ω1 and 0 in L(ω1+ω2) and
L(3ω1) and calculating the codimension of the eigenspace as above for L(ω3) does not
116
suffice either.
In the next lemma we consider a module that requires treatment for p = 2 only.
Lemma 6.6. Let G act on the irreducible module V = L(ω4) for n = 5 with p = 2. Then
(♦) holds for all X ∈ I.
Proof. We can use Lu¨beck’s tables [18] to see that the conclusion of Premet’s theorem
does not hold for this module, in particular, we have mω3 = mω1 = 0. In Figure 6.7 we
carry out the usual calculations to identify the weight strings for V and find that |Λ| > 48.
i ω |W.ω| mω
p = 2
4 ω4 80 1
3 ω3 80 0
2 ω2 40 2
1 ω1 10 0
0 0 1 4
Weight No. of l
strings strings p = 2
µ4 16
µ4 µ3 µ4 32 32
µ3 µ2 µ3 24
µ2 µ1 µ2 8 16
µ1 µ0 µ1 1
Lower bound on |Λ| 48
Figure 6.7: (λ, n) = (ω4, 5)
Thus we may certainly take two orthogonal roots, one short and one long, outside Φs.
Set x = 〈ω4, α〉 and y = 〈ω4, β〉. Then x can take the values 0 or ±2 and y can take
the values 0, ±1 or ±2. The weight strings of length three for y = 2 are of the form µ4
µ2 µ4. There are 12 pairs of roots with (x, y) = (2, 2), 24 with (x, y) = (2, 1), 24 with
(x, y) = (2, 0), 12 with (x, y) = (0, 2) and 24 with (x, y) = (0, 0). There are 240 pairs
of mutually orthogonal short and long roots in Φ. Thus, for a fixed pair α, β there are
80.12
240
= 4 weights µ ∈ W.ω4 with (x, y) = (2, 2). Taking the horizontal differences between
weights to be α and vertical β we see, therefore, that there are four 3 × 3 weight nets.
However, since the weights in W.ω3 do not appear for p = 2 we shall instead consider
these four nets as 2× 3 weight nets with a difference 2α between two adjacent horizontal
weights. Analogously, we find eight 2× 2, eight 2× 1 and four 1× 3 weight nets. There
are eight 1× 1 weight nets but these do not contribute to |Λ|. By considering weights in
W.ω2 we find two 2× 2 and eight 1× 2 weight nets. There is one 1× 3 weight net of the
form µ2 µ0 µ2. Therefore we have |Λ| > 4.5 + 8.2 + 8.1 + 4.2 + 2.4 + 1.4 + 8.2 = 80, so
(♦) holds.
117
We have not been able to draw any useful conclusions about the irreducible module
L(2ω1) with p 6= 2. The main difficulty is that dimL(2ω1) is only slightly larger than
dimG and most µ ∈ Π(2ω1) satisfy 〈µ, α〉 = 0 for a given short root α /∈ Φs. Thus most
weights lie in strings of length 1, so we can only show that |Λ| > 2n.
We next consider the spin module L(ωn) for n ∈ [7, 8]. We may label weights in W.ωn
by strings consisting of n plus and minus signs. This is done by considering the coefficients
of a weight µ ∈W.ωn, imagining a zero coefficient preceding the original coefficients of µ
and then placing a plus sign in position i of the string if the (i+ 1)st coefficient is larger
than the ith coefficient for i ∈ [0, n−1]; otherwise there is a minus sign. This simplifies the
task of determining whether the difference between two weights is a root and to determine
clusters. Two weights differ by αi+· · ·+αj for j < n if they are identical except a plus and
minus sign have been interchanged in the ith and (j + 1)st positions. Two weights differ
by αn if they are identical except one has a plus sign in the nth position of the string and
the other has a minus sign. It is clear, therefore, that if we compare two weights with one
having two plus signs and the other having two minus signs in the ith and jth positions
(otherwise identical) then they differ by the root αi+ · · ·+αj−1+2αj + · · ·+2αn. (Note
that an analogous method for labelling weights in W.ωn is available for types Cn when
p = 2 and Dn.)
Lemma 6.7. Let G act on the irreducible module L(ωn) for n ∈ [7, 8]. If n ∈ [7, 8] then
(♦) is satisfied for all X ∈ I.
Proof. It is clear that 〈ωn, α〉 = 1 for each of the n positive short roots α. Since |W (Bn) :
W (An)| = 2
n there are 2nn pairs (µ, α) for µ ∈ W.ωn and α ∈ ΦS with weight string
of the form µ µ − α. Thus, for a fixed short root α there are 2n−1 weight strings of the
form µ µ− α, i.e., all weights of L(ωn) appear in pairs differing by α. By the adjacency
principle we have |Λ| > 2n−1.
First assume n = 8; then we have |Λ| > 128, so (♦) holds unless X = ∅. Indeed, if
X = ∅ then we see that d∅ω8 > 129 > 128 = e
∅
ω8
by considering the clique below formed
from two pairs of weights with weights in a pair differing by α8. Note that we have omitted
118
a factor of 1
2
.
1 2 3 4 5 4 3 4, 1 2 3 4 5 4 3 2 and -1 0 1 2 3 2 1 2, -1 0 1 2 3 2 1 0
Now assume n = 7; then we have |Λ| > 64. Since |Φ(B7)| = 98, the condition (♦)
holds for all X ∈ I satisfying |Φ(X)| > 34. It remains to go through a list of 72 centraliser
types listed below. The calculations are routine.
Centraliser types
∅, A1, A21, A
3
1, D2, D2A1, D2A
2
1, B1, B1A1, B1A
2
1, B1A
3
1, B1A
2
1D2, B1A1D2, D2B1,
A2, A2A1, A2A
2
1, A2D2, A2D2A1, A2B1, A2B1A1, A2B1D2, A3, A3A1, A3D2, A3A2,
A3B1, A3B1A1, A3B1D2, A
2
2, A
2
2B1, A3D3, D3, D3A1, D3A
2
1, D3B1, D3A1B1, D3A2,
D3A2B1, A4, A4A1, A4D2, A5, A5B1, D4, D4A1, D4B1, D4A1B1, D4A2, B2, B2A1,
B2A
2
1, B2D2, B2D2A1, B2A2, B2A2A1, B2A2D2, B2A3, B2D3, B2D3A1, B2A4, B2D4,
B3, B3A1, B3A
2
1, B3D2, B3D2A1, B3A2, B3A3, B3D3, B4, B4A1
Table 6.2: Centraliser types requiring consideration for L(ω7) with n = 7
We remark that since all weights in W.ωn appear in pairs differing by a fixed short
root there is no benefit to taking further roots not in Φs and forming weight nets as we
have done in previous lemmas.
We shall now deal with the remaining modules which all have low rank.
Lemma 6.8. Let G act on the irreducible module V where we take V to be one of L(3ω2)
(p > 3), L(4ω2) (p > 3) or L(ω1 + 2ω2) (p 6= 2) with n = 2, or L(ω1 + ω3) with n = 3.
Then (†) holds for L(4ω2) with n = 2, L(ω1 + 2ω2) with n = 2 and p 6= 2, 3 and for
L(ω1 + ω3) with n = 3 and p 6= 7; otherwise (♦) is satisfied for all X ∈ I.
Proof. Consider the modules L(3ω2) and L(4ω2) for n = 2. In Figure 6.8 we show that
(†) holds and in Figure 6.9 we show that (♦) is satisfied for each X ∈ I since |Φ(B2)| = 8.
We consider the module L(ω1 + 2ω2) for n = 2 in Figure 6.10 below. We see that (†)
holds if p 6= 2, 3 and that (♦) is always satisfied if p = 3.
Next we consider the module L(ω1 + ω3) for n = 3. We see in Figure 6.11 that (†)
holds if p 6= 2, 7 and that (♦) is always satisfied if p = 7 since |Φ(B3)| = 18.
119
i ω |W.ω| mω
p 6= 2, 3
5 4ω2 4 1
4 ω1 + 2ω2 8 1
3 2ω1 4 1
2 2ω2 4 2
1 ω1 4 2
0 0 1 3
Weight No. of l
strings strings p 6= 2, 3
µ5 µ4 µ3 µ4 µ5 2 4
µ4 µ2 µ1 µ2 µ4 2 8
µ3 µ1 µ0 µ1 µ3 1 4
Lower bound on |Λ| 16
Figure 6.8: (λ, n) = (4ω2, 2)
i ω |W.ω| mω
p 6= 2, 3
3 3ω2 4 1
2 ω1 + ω2 8 1
1 ω2 4 2
Weight No. of l
strings strings p 6= 2, 3
µ3 µ2 µ2 µ3 2 4
µ2 µ1 µ1 µ2 2 6
Lower bound on |Λ| 10
Figure 6.9: (λ, n) = (3ω2, 2)
i ω |W.ω| mω
p 6= 2, 3 p = 3
4 ω1 + 2ω2 8 1 1
3 2ω1 4 1 1
2 2ω2 4 2 1
1 ω1 4 3 2
0 0 1 3 1
Weight No. of l
strings strings p 6= 2, 3 p = 3
µ4 µ3 µ4 2 2 2
µ4 µ2 µ1 µ2 µ4 2 8 4
µ3 µ1 µ0 µ1 µ3 1 5 3
Lower bound on |Λ| 15 9
Figure 6.10: (λ, n) = (ω1 + 2ω2, 2)
i ω |W.ω| mω
p 6= 7 p = 7
2 ω1 + ω3 24 1 1
1 ω3 8 3 2
Weight No. of l
strings strings p 6= 2, 7 p = 7 p = 2
µ2 µ2 8 8 8 8
µ2 µ1 µ1 µ2 4 16 12 20
Lower bound on |Λ| 24 20 28
Figure 6.11: (λ, n) = (ω1 + ω3, 3)
Lemma 6.9. Let G act on the irreducible module L(2ωn) (p 6= 2) for n ∈ [3, 4]. If n = 4
then (†) holds and if n = 3 then (♦) holds for all X ∈ I.
Proof. If n = 4 then Figure 6.12 shows that (†) certainly holds since dimG = 36.
If n = 3 then (♦) holds provided |Φ(X)| > 3. Therefore we need to consider separately
the possibilities X = ∅, A1 or B1.
We first note that W.ω1 consists of the short roots and W.ω2 the long roots hence it
is easy to identify in which Weyl group orbit a particular weight lies.
Suppose that X = ∅. If 000 ∈ Π(V ) \ Λ then no root lies in the eigenspace, i.e.,
Φ(B3) ⊂ Λ. Since there are 18 roots, 6 of which have multiplicity two (the short
roots), we have d∅2ω3 > 24 > 18 = e
∅
2ω3
. Now assume that 000 ∈ Λ. Suppose that
a long root lies in Π(V ) \ Λ, without loss of generality take 100. Then the weights
120
i ω |W.ω| mω
p 6= 2
4 2ω4 16 1
3 ω3 32 1
2 ω2 24 2
1 ω1 8 3
0 0 1 6
Weight No. of l
strings strings p 6= 2
µ4 µ3 µ4 8 8
µ3 µ2 µ3 12 24
µ2 µ1 µ2 6 18
µ1 µ0 µ1 1 6
Lower bound on |Λ| 56
Figure 6.12: (λ, n) = (2ω4, 4)
i ω |W.ω| mω
p 6= 2
3 2ω3 8 1
2 ω2 12 1
1 ω1 6 2
0 0 1 3
Weight No. of l
strings strings p 6= 2
µ3 µ2 µ3 4 4
µ2 µ1 µ2 4 8
µ1 µ0 µ1 1 3
Lower bound on |Λ| 15
Figure 6.13: (λ, n) = (2ω3, 3)
1 1 2, 1 1 1, 1 0 1, 1 1 0, 1 0 -1, 0 -1 0, 0 -1 -1 and 0 -1 -2
lie in Λ. We can arrange some of the remaining weights into the following three cliques:
1 2 2, 0 1 2, 0 1 1; 0 0 -1, -1 -1 -1, -1 -1 -2; and 1 2 3, 0 0 1.
Thus we have d∅2ω3 > 3 + 10 + 6 = 19. We now suppose that a short root lies in
Π(V ) \ Λ, without loss take this to be 0 0 1. Then the following weights lie in Λ:
1 2 3, 1 1 2, 0 1 2, 1 1 1, 1 0 1, 0 1 1, -1 0 1, 0 -1 0, 0 -1 -1, -1 -1 0, -1 -1 -1, -1 -2 -1;
so d∅2ω3 > 3 + 16 = 19.
Suppose that X = B1 with short root α3. The weights in Π(2ω3) can be arranged into
nine clusters as follows.
1 2 3 1 1 2 0 1 2 1 0 1 0 0 1 -1 0 1 0 -1 0 -1 -1 0 -1 -2 -1
1 2 2 1 1 1 0 1 1 1 0 0 0 0 0 -1 0 0 0 -1 -1 -1 -1 -1 -1 -2 -2
1 2 1 1 1 0 0 1 0 1 0 -1 0 0 -1 -1 0 -1 0 -1 -2 -1 -1 -2 -1 -2 -3
By reading from left to right, the 2nd, 3rd, 7th and 8th clusters have size 4 and the
5th cluster has size 7. If the 5th cluster lies in Λ then dB12ω2 > 28 since all other clusters
contain roots. Assume therefore that the 5th cluster lies in Π(V ) \ Λ. The 1st, 2nd and
3rd and the 7th, 8th and 9th clusters form two cliques so dB12ω2 > 7 + 14 = 21. Thus (♦)
holds.
Suppose that X = A1 with long root α1; here e
A1
2ω3
= 16. Consider the following seven
clusters.
1 1 2 1 1 1 1 0 1 1 0 0 1 0 -1 0 -1 -1 0 -1 -2
0 1 2 0 1 1 0 0 1 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -2
-1 0 1 -1 0 0 -1 0 -1
121
Assume that 000 ∈ Π(V ) \ Λ. Then all clusters above not containing 000 lie in Λ,
whence dA12ω3 > 20 and we are done. Instead let us assume that 000 ∈ Λ; we can form
three cliques from the 1st, 2nd and 3rd clusters and the 5th, 6th and 7th clusters. Thus
dA12ω3 > 5 + 6 + 6 = 17.
122
Chapter 7
Groups of type Cn
In this chapter we shall assume that G is a simple simply connected algebraic group of
type Cn defined over an algebraically closed field K and V = L(λ) is an irreducible G-
module with p-restricted highest weight λ. As in the previous chapter, it will be sensible
to treat the cases p 6= 2 and p = 2 separately since the conclusion of Premet’s theorem
does not hold in the latter case. We may assume that n > 3 since we studied B2(K) in
the previous chapter and B2(K) ∼= C2(K).
We shall prove the following result.
Theorem 7.1. Let G = Sp2n(K) act on V = L(λ). If dimV 6 dimG+ 2 then dimE =
dimV with the possible exceptions of L(ω2) for n ∈ [3,∞), L(ωn) for n ∈ [5, 6] and p = 2,
and L(ω3) for n = 3 and p 6= 2; if instead dimV > dimG + 2 then dimE < dimV with
the possible exceptions of L(ω4) with n = 4 and p 6= 2, and L(ω3) with n = 4 and p = 3.
This theorem is a consequence of the results to follow in later sections.
7.1 Initial survey
Assume throughout this section that p 6= 2. Consider µ 6 λ where µ =
∑n
i=1 aiωi is a
dominant weight. We shall begin by obtaining conditions on the coefficients ai in order
to show that (†) is satisfied for n large enough. This will allow us later to list modules
123
which will require further consideration. Recall that we may assume that there is a short
root outside the root system of the centraliser of a non-central semisimple element and in
calculations to follow we shall be considering short roots only: see Section 4.3.
It is useful to note that ωk−2 < ωk for k > 3; this is clear since each fundamental
weight is of the form ωi = α1+2α2+ · · ·+(i−1)αi−1+ i
(
αi + · · ·+ αn−1 +
1
2
αn
)
for each
i ∈ [1, n].
Proposition 7.2. Suppose that µ =
∑n
i=1 aiωi 6 λ is a dominant weight, and at least
one of the following conditions holds:
(i) n > 9 and ak 6= 0 for some k ∈ [3, n];
(ii) n ∈ [7, 8] and ak 6= 0 for some k ∈ [4, n];
(iii) n = 6 and either a4 6= 0 or a6 6= 0;
(iv) n > 5 and ai, aj 6= 0 for some i ∈ [1, 3] and j ∈ [n− 1, n];
(v) n = 4 and ai, a3 6= 0 for some i ∈ [1, 2].
Then (†) holds.
Proof. Assume that ak 6= 0 for some k ∈ [3, n]. Then Ψ = 〈αi | ai = 0〉 is contained in
Φ(Ak−1Cn−k) and
rΨ > rAk−1Cn−k = 2
k−2k
(
n
k
)
4n− 3k − 1
n(n− 1)
.
If k = 4 then rA3Cn−4 > dimG = n(2n + 1) for n > 6 and rA3Cn−4 6 rAk−1Cn−k for
k ∈ [4, n− 2]. Thus (†) holds for n > 6 provided that ak 6= 0 for k ∈ [4, n− 2]. Also, by
taking k = 3 and k = n−1, we find that rA2Cn−3 > dimG if n > 9 and rAn−2C1 > dimG if
n > 7. If k = n then rAn−1 > dimG whenever n > 10; moreover, if n ∈ [6, 9] and an 6= 0
then (†) holds by Premet’s theorem since ωn−2 < ωn.
Suppose that both a1 6= 0 and an 6= 0. Then Ψ ⊂ Φ(An−2) and rΨ > rAn−2 =
2n−2(n + 2) exceeds dimG for n > 5. Furthermore, since rAn−2 6 rA1An−3 6 rA2An−4 , we
cannot have an 6= 0 and either a2 or a3 non-zero.
124
We calculate rA2An−5C1 =
2n−4
3
(n − 2)(n − 3)(n2 + 7n − 26), rA1An−4C1 = 2
n−4(n −
2)(n2 + 5n − 14) and rAn−3C1 = 2
n−3(n2 + 3n − 6) so that (†) holds if both a3 and an−1
are non-zero for n > 5 and if both ai and an−1 are non-zero with i ∈ [1, 2] for n > 4.
The proposition above very much restricts the possible highest weights for our irre-
ducible modules.
The modules that have dimension at most dimG+2 are as follows: L(ω1), L(ω2) and
the adjoint module L(2ω1) for n > 3, and the module L(ω3) for n = 3. Only L(ω1) for
n > 3 has dimension less than dimB = n(n+1) so we are left to consider L(ω2) for n > 3
and L(ω3) for n = 3.
Lemma 7.3. Suppose that dimV > dimG + 2. If n > 3 then (†) holds except possibly
for the modules with highest weight ω4 for n ∈ [4, 5], ω1 + ω3 for n = 3, ω5 for n = 5, ω3
for n ∈ [4, 8], and both ω1 + ω2 and 3ω1 (p > 3) for n ∈ [3, 4].
Proof. We shall split the analysis of weights to follow into six parts according as n > 9,
n ∈ [7, 8] or n = 6, 5, 4 or 3.
Case I: n > 9. By condition (i) of Proposition 7.2 we may assume that µ = a1ω1+a2ω2.
Let us set m1 =
∑2
i=1 iai so that, as in Lemma 5.3, if m1 > 3 then for some bi with
0 6 bi 6 ai for each i ∈ [1, 2] we have
µ′ = µ−
(
2∑
i=1
biωi − ωj
)
< µ 6 λ,
where j =
∑2
i=1 ibi and 3 6 j 6 4. Since the coefficient a
′
j of µ
′ is non-zero, by applying
Premet’s theorem, we may assume that m1 6 2. Hence the possible weights µ correspond
to modules with dimension at most dimG.
Case II: n ∈ [7, 8]. By condition (ii) of Proposition 7.2, we may assume that ak = 0
for k ∈ [4, n], whence µ =
∑3
i=1 aiωi. Set M1 =
∑3
i=1 iai; if M1 > 4 then, as in Lemma
5.3 there exist bi for each i ∈ [1, 3] satisfying 0 6 bi 6 ai such that
µ′ = µ−
(
3∑
i=1
biωi − ωj
)
< µ 6 λ,
125
where j =
∑3
i=1 ibi and 4 6 j 6 6. Since a
′
j 6= 0 in µ
′ we may assume, by Premet’s
theorem, that M1 6 3.
Considering the weight ω1 + ω2 we calculate rA2Cn−3 = 2(n − 2)(2n − 5) and rCn−2 =
4(2n − 3) since ω1 + ω2 > ω3. Thus we have |Λ| > 44 + 90 = 134 for n = 7 and
|Λ| > 52 + 132 = 184 for n = 8, and (†) holds in both cases. Therefore we must consider
further the weight ω3 for n ∈ [7, 8].
Case III: n = 6. By conditions (iii) and (iv) of Proposition 7.2 we may assume that
µ =
∑3
i=1 aiωi + a5ω5 and that we do not have both ai 6= 0 and a5 6= 0 for i ∈ [1, 3].
Thus, if a5 6= 0 then µ = a5ω5. In fact (†) always holds in this case since ω5 > ω3 and
rA4C1+rA2C3 = 64+56 = 120 > dimG. We may now assume that µ =
∑3
i=1 aiωi. Suppose
first that precisely two coefficients of µ are non-zero. The condition (†) holds if a1 and
a3 are non-zero since ω1 + ω3 > ω4, and if a1 and a2 are non-zero since ω1 + ω2 > ω3 and
rC4 + rA2C3 = 36 + 56 = 92. Similarly if a2 and a3 are non-zero since ω2 + ω3 > ω1 + ω2.
In particular this shows that at most one coefficient of µ can be non-zero. We can use
Premet’s theorem and the partial orderings 3ω1 > ω1 + ω2, 2ω2 > ω4 and 2ω3 > ω1 + ω3
to deduce that we need only to consider further the weight ω3 for n = 6.
Case IV: n = 5. By condition (iv) of Proposition 7.2 we may assume that we do not
have both ai 6= 0 and aj 6= 0 for i ∈ [1, 3] and j ∈ [4, 5]. If both a4 and a5 are non-zero
then (†) holds since rA3 = 56. If a4 6= 0 then we may assume a4 = 1 since 2ω4 > ω2 + ω4.
Similarly, if a5 6= 0 then we may assume a5 = 1 since 2ω5 > ω3 + ω5.
We may now assume that µ =
∑3
i=1 aiωi. If both a2 and a3 are non-zero then (†)
holds since rA1C2 = 102; the same is true if both a1 and a3 are non-zero. If both a1
and a2 are non-zero then (†) holds by Premet’s theorem since ω1 + ω2 > ω3 and we have
rC3 + rA2C2 = 28 + 30 = 58. Thus we can assume that precisely one coefficient of µ is
non-zero. We must have a1 6 2 since 3ω1 > ω1 + ω2. Also if a2 6= 0 then a2 = 1 since
2ω2 > ω1+ω3 and if a3 6= 0 then a3 = 1 since 2ω3 > ω2+ω4. Therefore we must consider
further ω3, ω4 and ω5 for n = 5.
Case V: n = 4. By condition (v) of Proposition 7.2 we see that (†) holds if both ai 6= 0
and a3 6= 0 for i ∈ [1, 2]. Suppose that both a1 and a4 are non-zero. Then Ψ ⊂ Φ(A2)
126
and rA2 = 24. We see that (†) holds since rC2 = 20 and ω1 + ω4 > ω1 + ω2. Similarly we
are done if both a2 and a4 are non-zero since then Ψ ⊂ Φ(A
2
1) and rA21 = 40. Therefore
we can have at most two coefficients of µ non-zero and if two coefficients are non-zero
then either a1 6= 0 and a2 6= 0 or a3 6= 0 and a4 6= 0. The condition (†) holds in the
latter case since ω3 + ω4 > ω2 + ω3. In the former case we must have a1 = a2 = 1 since
2ω1 + ω2 > ω1 + ω3 and ω1 + 2ω2 > ω2 + ω3.
Assume that precisely one coefficient of µ is non-zero. If a1 > 4 then (†) holds since
4ω1 > ω1+ω3. Similarly if ai > 2 for i ∈ [2, 4] since 2ω4 > 2ω3 > 2ω2 > ω1+ω3. Therefore
we must consider further ω1 + ω2, 3ω1, ω3 and ω4.
Case VI: n = 3. If both a2 and a3 are non-zero then (†) holds by Premet since
ω2 + ω3 > ω1 + ω2 and we have rA1 = 10 and rC1 = 12. In particular we see that at most
two coefficients of µ can be non-zero. If both a1 and a3 are non-zero then (†) holds if
a1+ a3 > 2 since ω1+2ω3 > ω2+ω3 and 2ω1+ω3 > ω2+ω3. The weight ω1+ω3 satisfies
ω1+ω3 > 2ω1 > ω2 but rA1 + rC2 + rA1C1 = 10+2+5 = 17 which does not exceed dimG.
If both a1 and a2 are non-zero then (†) holds if a1 + a2 > 2 since ω1 + 2ω2 > ω2 + ω3
and since 2ω1 + ω2 > ω1 + ω3. Now assuming that only one coefficient of µ is non-zero
we may take a1 6 3 since 4ω1 > 2ω1 + ω2 in which case (†) holds, a2 = 1 by above since
2ω2 > ω1 + ω3 and a3 = 0 since 2ω3 > 2ω2 and L(ω3) has dimension less than dimG. We
must consider further ω1 + ω2, ω1 + ω3 and 3ω1.
7.2 Even characteristic
Assume now that p = 2. By Zalesski’s Theorem 2.7 and Theorem 2.8 any 2-restricted
weight with short support can be treated in the same way as the previous section since
the conclusion of Premet’s theorem holds true and the associated module is tensor inde-
composable. Thus (†) holds for all 2-restricted modules with short support except those
listed in Lemma 7.3.
The tensor decomposable modules take the form L
(∑n−1
i=1 aiωi
)
⊗ L(ωn) where each
ai ∈ {0, 1}. Note that this module is isomorphic to L
(∑n−1
i=1 aiωi + ωn
)
by Theorem 2.4.
127
We shall use Proposition 4.3 to show that (†) holds for these modules. In Lemma 7.3 we
have shown that (†) holds for all but a short list of modules (most of which are 2-restricted
with short support). Therefore, after applying Proposition 4.3 it remains to investigate
the modules L(µ)⊗L(ωn) where µ is one of ω4 for n = 5, ω3 for n ∈ [4, 8], and ω1+ω2 for
n ∈ [3, 4] (the relevant exceptions in Lemma 7.3), as well as ω1 and ω2 both for n ∈ [3,∞)
(since L(ω1) and L(ω2) have dimensions at most dimG).
Lemma 7.4. If G = Sp2n(K) acts on a tensor decomposable p-restricted module V then
(†) holds except possibly if V = L(ω1)⊗ L(ωn) for n ∈ [3, 4].
Proof. We shall apply Proposition 4.3 to each of the tensor decomposable modules. First
we notice that, in any case, we may assume n 6 6 since dimL(ωn) > dimG otherwise.
Consider L(ω1 + ω2) ⊗ L(ωn) for n ∈ [3, 4]. Recall that the conclusion of Premet’s
theorem holds for ω1 + ω2 ∈ X(T )S. We find that
|Λω1+ω2 | > rCn−2 =
1
2
2nn!
2n−2(n− 2)!
2n(n− 1)− 2(n− 2)(n− 3)
2n(n− 1)
= 4(2n− 3),
whence |Λω1+ω2| dimL(ωn) > 2
n+2(2n− 3); this exceeds dimG for n ∈ [3, 4].
Consider L(ω4) ⊗ L(ω5) for n = 5. Since |Λω4| > rA3C1 = 28 and dimL(ω5) = 32 we
have that (†) holds.
Consider L(ω2) ⊗ L(ωn) for n ∈ [3, 6]. We find that |Λω2 | > rA1Cn−2 = 4n − 7 so
|Λω2 | dimL(ωn) > dimG for n ∈ [3, 6]. We deal with L(ω3) ⊗ L(ωn) for n ∈ [4, 6]
analogously since rA2Cn−3 = 2(n− 2)(2n− 5) > rA1Cn−2 .
Finally, consider the module L(ω1) ⊗ L(ωn) for n ∈ [3, 6]. We have |Λωn| > rAn−1 =
2n−2, whence |Λωn| dimL(ω1) > 2
n−1n exceeds dimG for n > 5.
We must examine L(ωn) for p = 2 and n ∈ [7,∞); this module has dimension 2
n.
We have 〈ωn, α〉 = 0 or ±2 for short roots α ∈ Φ(Cn) and in particular, 〈ωn, α〉 = 2 for
1
2
n(n− 1) short roots. Thus, for a fixed short root α there are 2
n−1n(n−1)
2n(n−1)
= 2n−2 weights
µ ∈ W.ωn with weight string containing µ µ + 2α. Since p = 2 we obtain |Λ| > 2
n−2
by the adjacency principle; this exceeds dimG for n > 10. Hence, in the next section, it
128
remains for us to consider L(ωn) for p = 2 and n ∈ [7, 9].
7.3 Weight string analysis
We list in Table 7.1 the weights which require further consideration.
λ n Lemma
ω4 5 7.6 (†) (p 6= 3), (♦) (p = 3)
4 (p 6= 2) 7.6 (♦)a
ω1 + ωn 4 (p = 2) 7.5 (†)
3 7.5 (†)
ω5 5 (p 6= 2) 7.6 (♦)
ω3 [7, 8] 7.7 (†)
[5, 6] 7.7 (♦)
4 7.7 (♦)b
ω1 + ω2 [3, 4] 7.5 (†)
3ω1 [3, 4] (p 6= 2, 3) 7.5 (†)
ωn [7, 9] (p = 2) 7.8 (♦)
Except for: aX = A21, A3 and C
2
2 ,
bX = A3 if p = 3.
Table 7.1: Possible weights in type Cn for n > 3
Lemma 7.5. Let G act on the irreducible module V where we take V to be one of L(ω1+
ω3) for n = 3, L(ω1 + ω4) for n = 4 and p = 2, L(3ω1) for n ∈ [3, 4] and p > 3, or
L(ω1 + ω2) for n ∈ [3, 4]. Then (†) holds in each case.
Proof. Consider L(ω1 + ω3) with n = 3. In the last column of the second table in Figure
7.1 we are using fact that if p = 2 then 2α(s) = 1 if and only if α(s) = 1. We see that (†)
holds for all p.
Now consider L(ω1 + ω4) for n = 4 and p = 2. By ignoring the weights in W.ω1 and
W.(ω1 + ω2) (since they occur with multiplicity zero) we see from Figure 7.2 below that
(†) holds.
Consider the module L(3ω1) for n ∈ [3, 4]; we have 3ω1 > ω1 + ω2 > ω3 > ω1. It is
more convenient to take these two low rank cases separately since certain weight strings
simply do not occur for n = 3.
129
i ω |W.ω| mω
p 6= 2, 3 p = 3 p = 2
3 ω1 + ω3 24 1 1 1
2 2ω1 6 1 1 0
1 ω2 12 3 2 2
0 0 1 4 3 0
Weight No. of l
strings strings p 6= 2, 3 p = 3 p = 2
µ3 4
µ3 µ3 4 4 4 4
µ3 µ2 µ3 2 2 2 2
µ3 µ1 µ1 µ3 4 16 12 16
µ2 µ2 2 2 2
µ1 2
µ1 µ0 µ1 1 4 3 2
Lower bound on |Λ| 28 23 24
Figure 7.1: (λ, n) = (ω1 + ω3, 3)
i ω |W.ω| mω
p = 2
4 ω1 + ω4 64 1
3 ω1 + ω2 48 0
2 ω3 32 2
1 ω1 8 0
Weight No. of l
strings strings p = 2
µ4 16
µ4 µ4 8 8
µ4 µ3 µ4 8 8
µ4 µ2 µ2 µ4 8 32
µ3 µ3 10
µ3 µ2 µ3 8
µ3 µ1 µ1 µ3 2
µ2 µ1 µ2 4 8
Lower bound on |Λ| 56
Figure 7.2: (λ, n) = (ω1 + ω4, 4)
If n = 4 the usual calculations with weight strings shows that |Λ| > 2rC3+rC2+rA2C1 =
4 + 20 + 12 = 36 = dimG. We show in Figure 7.3 that (†) holds by using weight
multiplicities.
i ω |W.ω| mω
p 6= 2, 3
4 3ω1 8 1
3 ω1 + ω2 48 1
2 ω3 32 1
1 ω1 8 4
Weight No. of l
strings strings p 6= 2, 3
µ4 4
µ4 µ3 µ3 µ4 2 4
µ3 8
µ3 µ3 8 8
µ3 µ2 µ3 8 8
µ3 µ1 µ1 µ3 2 10
µ2 µ2 8 8
µ2 µ1 µ2 4 8
Lower bound on |Λ| 46
Figure 7.3: (λ, n) = (3ω1, 4)
Similarly, if n = 3 we have |Λ| > 2rC2 + rC1 + rA2 = 4+ 12+ 2 = 18 = |Φ(G)|, and we
show in Figure 7.4 that (†) holds by using weight multiplicities.
Consider L(ω1+ω2) for n ∈ [3, 4]. In Figures 7.5 and 7.6 for n = 4 and n = 3 we show
that (†) holds for all characteristics.
Recall from [2, p.47] that for the root system of type Cn we have Φ = ΦS ∪ΦL where
ΦS = {±ǫi ± ǫj | 1 6 i < j 6 n} consists of 2n(n− 1) short roots and ΦL = {±2ǫi | 1 6
130
i ω |W.ω| mω
p 6= 2, 3
4 3ω1 6 1
3 ω1 + ω2 24 1
2 ω3 8 1
1 ω1 6 3
Weight No. of l
strings strings p 6= 2, 3
µ4 2
µ4 µ3 µ3 µ4 2 4
µ3 µ3 4 4
µ3 µ2 µ3 4 4
µ3 µ1 µ1 µ3 2 8
µ2 µ1 µ2 2 4
Lower bound on |Λ| 24
Figure 7.4: (λ, n) = (3ω1, 3)
i ω |W.ω| mω
p 6= 3 p = 3
3 ω1 + ω2 48 1 1
2 ω3 32 2 1
1 ω1 8 6 4
Weight No. of l
strings strings p 6= 2, 3 p = 3 p = 2
µ3 8
µ3 µ3 10 10 10 10
µ3 µ2 µ3 8 16 8 16
µ3 µ1 µ1 µ3 2 14 10 16
µ2 µ2 8 16 8 16
µ2 µ1 µ2 4 16 8 16
Lower bound on |Λ| 72 44 74
Figure 7.5: (λ, n) = (ω1 + ω2, 4)
i 6 n} consists of 2n long roots. Let αi = ǫi − ǫi+1 for i ∈ [1, n− 1] and αn = 2ǫn be the
simple roots. Therefore we have ωi =
∑i
k=1 ǫk. We shall need this for the next lemma.
Lemma 7.6. Let G act on the irreducible module V where we take V to be one of L(ω5)
for n = 5 with p 6= 2, L(ω4) for n = 4 with p 6= 2 or L(ω4) for n = 5. If V = L(ω4) for
n = 5 with p 6= 3 then (†) holds. If V is either L(ω5) for n = 5 with p 6= 2 or L(ω4) for
n = 5 with p = 3 then (♦) is satisfied for all X ∈ I. If V = L(ω4) for n = 4 with p 6= 2
then (♦) holds for all X ∈ I \ {A21, A3, C
2
2}.
Proof. Consider the module L(ω4) for n = 5. In Figure 7.7 we see that (†) holds unless
p = 3.
If p = 3 then (♦) holds for each X ∈ I such that |Φ(X)| > 9. We may therefore
assume that there are two orthogonal roots, α (short) and β (long), outside Φs. It is
straightforward to check that 〈ω4, α〉 = 0, ±1 or ±2 and 〈ω4, β〉 = 0 or ±1. We note that
for a root system of type Cn there are 2n.2(n−1)(n−2) orthogonal pairs of roots α short,
β long, so 240 such pairs if n = 5.
We have 〈ω4, α〉 = 2 and 〈ω4, β〉 = 1 for 12 pairs of roots α, β with (α, β) = 0. In more
detail, the 12 pairs of orthogonal long and short roots are 2ǫi and ǫj+ǫk with i, j, k ∈ [1, 4],
j < k and k 6= i 6= j. There are 80.12 = 960 triples (µ;α, β) with 〈µ, α〉 = 2, 〈µ, β〉 = 1
131
i ω |W.ω| mω
p 6= 3, 7 p = 7 p = 3
3 ω1 + ω2 24 1 1 1
2 ω3 8 2 2 1
1 ω1 6 4 3 3
Weight No. of l
strings strings p 6= 2, 3, 7 p = 7 p = 3 p = 2
µ3 µ3 6 6 6 6 6
µ3 µ2 µ3 4 8 8 4 8
µ3 µ1 µ1 µ3 2 10 8 8 12
µ2 µ1 µ2 2 8 8 4 8
Lower bound on |Λ| 32 28 22 34
Figure 7.6: (λ, n) = (ω1 + ω2, 3)
i ω |W.ω| mω
p 6= 2, 3 p = 2 p = 3
2 ω4 80 1 1 1
1 ω2 40 2 2 1
0 0 1 5 4 1
Weight No. of l
strings strings p 6= 2, 3 p = 2 p = 3
µ2 24
µ2 µ2 16 16 16 16
µ2 µ1 µ2 12 24 24 12
µ1 2
µ1 µ1 12 24 24 12
µ1 µ0 µ1 1 4 4 1
Lower bound on |Λ| 68 68 41
Figure 7.7: (λ, n) = (ω4, 5)
for orthogonal roots α ∈ ΦS, β ∈ ΦL and µ ∈ W.ω4. For a given pair α, β there are
960
240
= 4 weights µ ∈ W.ω4 such that 〈µ, α〉 = 2 and 〈µ, β〉 = 1. Thus there are four 3× 2
weight nets and similarly we calculate that there are four 3×1, eight 2×2 and eight 1×2
weight nets. We cannot obtain 2× 1 weight nets since 〈ω4, α〉 = 1 and 〈ω4, β〉 = 0 is not
possible and we need not consider weights µ ∈ W.ω4 with 〈µ, α〉 = 〈µ, β〉 = 0 since there
is no contribution to |Λ| from 1×1 weight nets. So from weight nets containing µ ∈W.ω4
we find that |Λ| > 4.3 + 4.1 + 8.2 + 8.1 = 40.
Now we consider the remaining weight nets for L(ω4), i.e., those containing ν ∈W.ω2
and no weights from W.ω4. As before, we can check that 〈ω2, α〉 = 0, ±1 or ±2 and
〈ω2, β〉 = 0 or ±1. It is not possible to find pairs of orthogonal roots α and β with
〈ω2, α〉 = 2 and 〈ω2, β〉 = 1. We find that there are two 2 × 2 and eight 2 × 1 weight
nets and there is one 3 × 1 weight net (which has the form µ1 µ0 µ1). The contribution
to |Λ| from the weight nets formed from weights ν ∈W.ω2 with 〈ν, α〉 = 0 and 〈ν, β〉 = 1
was counted earlier since these nets appear in the 3 × 2 weight nets above. Therefore
|Λ| > 40 + 2.2 + 8.1 + 1.1 = 53 and (♦) is satisfied for all X ∈ I.
132
Consider the module L(ω5) for n = 5. When p = 2 the module has dimension less
than dimG, so we assume that p 6= 2. There are three dominant weight orbits for this
module; they are related as follows: ω5 > ω3 > ω1. If p 6= 2, 3 then mω5 = mω3 = 1 and
mω1 = 2 whereas if p = 3 then mν = 1 for each ν ∈ Π(ω5). Thus in the latter case we do
not gain anything from considering weight multiplicities. We use Proposition 4.3 to see
that |Λ| > rA4 + rA2C2 + rC4 = 8+30+2 = 40. The condition (♦) is satisfied for all X ∈ I
with |Φ(X)| > 10 and it is clear that we may assume that there are two orthogonal roots
(one long and one short) not in Φs.
As before we may calculate the number of weight nets, however we arrange the weights
in Π(ω5) into weight nets provided explicitly below (ordered by nets of the same size: 1×2,
2×2, 3×2, 2×1, 3×1 and one 3×2 weight net that does not have a negative counterpart
since it is unchanged under negation) where the difference between horizontal weights is
α1 and vertical weights is α5. We omit the eight 1× 1 weight nets.
1 2 3 4 5
2
1 2 3 2 3
2
1 2 2 2 3
2
1 2 1 2 3
2
1 2 1 0 1
2
1 2 3 4 3
2
1 2 3 2 1
2
1 2 2 2 1
2
1 2 1 2 1
2
1 2 1 0 - 1
2
1 1 2 2 3
2
0 1 2 2 3
2
1 1 1 2 3
2
0 1 1 2 3
2
1 1 1 0 1
2
0 1 1 0 1
2
1 1 0 0 1
2
0 1 0 0 1
2
1 1 2 2 1
2
0 1 2 2 1
2
1 1 1 2 1
2
0 1 1 2 1
2
1 1 1 0 - 1
2
0 1 1 0 - 1
2
1 1 0 0 - 1
2
0 1 0 0 - 1
2
1 0 1 2 3
2
0 0 1 2 3
2
-1 0 1 2 3
2
1 0 1 0 1
2
0 0 1 0 1
2
-1 0 1 0 1
2
1 0 1 2 1
2
0 0 1 2 1
2
-1 0 1 2 1
2
1 0 1 0 - 1
2
0 0 1 0 - 1
2
-1 0 1 0 - 1
2
1 1 2 3 3
2
0 1 2 3 3
2
1 1 2 1 1
2
0 1 2 1 1
2
1 1 1 1 1
2
0 1 1 1 1
2
1 1 0 1 1
2
0 1 0 1 1
2
1 1 0 -1 - 1
2
0 1 0 -1 - 1
2
1 0 1 1 1
2
0 0 1 1 1
2
-1 0 1 1 1
2
1 0 0 1 1
2
0 0 0 1 1
2
-1 0 0 1 1
2
1 0 0 0 1
2
0 0 0 0 1
2
-1 0 0 0 1
2
1 0 0 0 - 1
2
0 0 0 0 - 1
2
-1 0 0 0 - 1
2
If we take the weight nets above together with the corresponding negative versions for
all but the last weight net then, by the adjacency principle, we have |Λ| > 55 if p = 3 and
|Λ| > 62 if p 6= 2, 3. Thus (♦) is satisfied for all X ∈ I.
Consider the module L(ω4) for n = 4. If p = 2 then dimL(ω4) = 16 < 20 = dimB
so we may assume that p 6= 2. We may use the available information about weight
multiplicities to show that |Λ| > 13 or 14 according as p = 3 or p 6= 2, 3. As usual (♦) is
satisfied for X ∈ I satisfying |Φ(X)| > 18 or 19; the remaining centraliser types are ∅, A1,
A21, A2, A3, C1, C1A1, C1A2, C
2
1 , C
2
1A1, C2, C2A1, C2C1, C
2
2 and C3. It is straightforward
133
to show that (♦) holds for all but three of these. If X is one of A3, C
2
2 or A
2
1 then we may
have equality in (♦) so this module for p 6= 2 is a possible exception in Theorem 7.1.
Lemma 7.7. Let G act on the irreducible module L(ω3) for n ∈ [4, 8]. If n ∈ [7, 8] then
(†) holds and if n ∈ [5, 6] or n = 4 with p 6= 3 then (♦) holds for each X ∈ I. If n = 4
with p = 3 then (♦) holds for all X ∈ I \ {A3}.
Proof. Consider the module L(ω3) with n ∈ [4, 8]. It is necessary to separate the case
n = 4 = p+1 from the others in the right-hand table of Figure 7.8 because mω1 = 1 here.
We see that for n ∈ [7, 8] the condition (†) holds.
i ω |W.ω| mω
p ∤ n− 1 p | n− 1
2 ω3
4
3
n(n− 1)(n− 2) 1 1
1 ω1 2n n− 2 n− 3
Weight No. of l
strings strings p ∤ (n− 1) p | (n− 1) 6= 3 p | (n− 1) = 3
µ2
4
3
(n− 2)(n2 − 7n+ 15)
µ2 µ2 4(n− 2)(n− 3) 4(n− 2)(n− 3) 4(n− 2)(n− 3) 8
µ2 µ1 µ2 2(n− 2) 4(n− 2) 4(n− 2) 4
µ1 µ1 2 2(n− 2) 2(n− 3) 2
Lower bound on |Λ| 2(n− 2)(2n− 3) 2(n− 1)(2n− 5) 14
Figure 7.8: λ = ω3 for n ∈ [4, 8]
Suppose that we can find two orthogonal roots, α (short) and β (long), outside Φs.
We see that 〈ω3, α〉 = 0, ±1 or ±2 and 〈ω3, β〉 = 0 or ±1. We note that for a root system
of type Cn there are 4n(n− 1)(n− 2) orthogonal pairs of roots α short and β long.
We have 〈ω3, α〉 = 2 and 〈ω3, β〉 = 1 for 3 pairs of roots α, β with (α, β) = 0. These
3 pairs of orthogonal long and short roots are ǫ2 + ǫ3 and 2ǫ1, ǫ1 + ǫ3 and 2ǫ2, and 2ǫ3
and ǫ1 + ǫ2. There are
4
3
n(n− 1)(n− 2).3 triples (µ;α, β) with 〈µ, α〉 = 2, 〈µ, β〉 = 1 for
orthogonal roots α ∈ ΦS, β ∈ ΦL and µ ∈ W.ω3. For a given pair α, β there is one weight
µ ∈ W.ω4 such that 〈µ, α〉 = 2 and 〈µ, β〉 = 1. Thus there is one 3 × 2 weight net and
similarly we calculate that there are 2(n− 3) 3× 1, 4(n− 3) 2× 2, 4(n− 3)(n− 4) 2× 1
and 2 + 2(n− 3)(n− 4) 1× 2 weight nets.
Now we consider the remaining weight nets for L(ω3), i.e., those containing ν ∈W.ω1
and no weights fromW.ω3. As before, we can check that 〈ω1, α〉 = 0 or ±1 and 〈ω1, β〉 = 0
134
or ±1. It is not possible to find pairs of orthogonal roots α and β with 〈ω1, α〉 = 〈ω1, β〉 =
1. We find that there are two 2× 1 weight nets. The contribution to |Λ| from the weight
net formed from weights ν ∈ W.ω1 with 〈ν, α〉 = 0 and 〈ν, β〉 = 1 was counted earlier
since this net appears in the 3× 2 weight net above.
If n = 6 then from Figure 7.8 we see that |Λ| > 72 or 70 according as p 6= 5 or p = 5.
There are certainly two orthogonal roots, one short and one long, outside Φs. Therefore,
if p = 5 we have |Λ| > 14 + 24 + 24 + 12 + 5 + 6 = 85. This exceeds dimG and the same
is true for p 6= 5 since the weights in W.ω1 have larger multiplicity.
If n = 5 we see from Figure 7.8 that |Λ| > 40 or 42 according as p = 2 or p 6= 2 and if
n = 4 then |Λ| > 14 or 20 according as p = 3 or p 6= 3. Thus (♦) is satisfied for n = 5 if
|Φ(X)| > 8 or 10 according as p 6= 2 or p = 2 and for n = 4 if |Φ(X)| > 12 or 18 according
as p 6= 3 or p = 3. In fact if n = 4 and p = 3 then all X ∈ I are possible except C3C1 and
C4.
We provide the subsystems X in Table 7.2 below that remain to be dealt with to show
that (♦) is satisfied for all X ∈ I. For n = 5 there are three subsystems which need
consideration for p = 2 only; these are presented at the end of the n = 5 row. Similarly
with n = 4 there are two subsystems to be treated for p = 3 only.
n Centraliser type
5 ∅, A1, C1, A21, A1C1, C
2
1 , A1C
2
1 , A
2
1C1, A2, A2A1, A2C1, C2; A2C
2
1 , C2A1, C2C1
4 ∅, A1, C1, A21, A1C1, C
2
1 , A1C
2
1 , A2, A2C1, C2, C2A1, C2C1, A3; C
2
2 , C3
Table 7.2: Centraliser types requiring consideration for L(ω3) with n ∈ [4, 5]
If n = 5 we may assume that there are two orthogonal roots, α (short) and β (long),
outside Φs. Using the weight nets calculated above we find that |Λ| > 49 or 47 according
as p 6= 2 or p = 2. We see that (♦) is satisfied unless X = ∅ (for all p), or A1 or C1 for
p = 2.
We may take three orthogonal roots outside Φs, two short roots and one long root as
in Lemma 7.6. Set x = 〈ω3, α〉, y = 〈ω3, β〉 and z = 〈ω3, γ〉 where α and γ are short
roots and β is a long root. Note that there are the same number of triples of roots α,
β, γ taking values (a, b, c) and (c, b, a). We find that there are 12 triples of such roots
135
with (x, y, z) = (2, 1, 0), 48 with (x, y, z) = (1, 1, 1), 48 with (x, y, z) = (0, 1, 0), 24 with
(x, y, z) = (2, 0, 1) and 48 with (x, y, z) = (1, 0, 0). Since |W.ω3| = 80 and there are 960
triples of two short roots and one long root (mutually orthogonal) we calculate that, for
a given triple α, β, γ there is one weight µ ∈ W.ω3 with 3× 2× 1 weight net, one weight
with 1 × 2 × 3 weight net, four weights with 2 × 2 × 2 weight nets, four weights with
1× 2× 1 weight net, two weights with 3× 1× 2 weight net, two weights with 2× 1× 3
weight net, four weights with 2× 1× 1 weight net and four weights with 1× 1× 2 weight
net. Note that weight nets with side of length 3 here consist of weight strings of the form
µ2 µ1 µ2.
Since 〈ω1, α〉 = 〈ω1, γ〉 = 0 and 〈ω1, β〉 = 1 for 96 triples of roots α, β, γ andW.ω1 = 10
we see, by taking such a triple, that there is only one weight with 1 × 2 × 1 weight net.
This means that the string µ1 µ1 (where the weights in this string differ by β) in the
3 × 2 × 1 and the 1 × 2 × 3 weight net are the same. These should be combined to
form a three-dimensional plus shape consisting of eight weights in W.ω3 occurring with
multiplicity 1 and two weights in W.ω1 occurring with multiplicity 3 or 2 according as
p 6= 2 or p = 2.
If we take α, β, γ /∈ Φs then we have |Λ| > 7 or 8 from the plus-shaped net according
as p 6= 2 or p = 2 and |Λ| > 10 for each of the 3× 1× 2 and 2× 1× 3 weight nets for any
p. Thus |Λ| > 55 or 56 according as p 6= 2 or p = 2. In both cases (♦) is satisfied for all
X ∈ I.
If n = 4 with p 6= 3 then we can take two orthogonal roots outside Φs as before to
show that |Λ| > 22, which is a small improvement. If p = 3 we need to deal with the cases
X = C22 and C3 before we can assume that there are two orthogonal roots, one short and
one long, not in Φs. It is straightforward to run through each of the possibilities for n = 4
in Table 7.2 with p 6= 3 to show that (♦) is satisfied. However, if p = 3 it is only possible
to deal with all X ∈ I \ {A3}. Suppose that X = A3 with simple roots α1, α2 and α3.
There are four cliques and weights in a clique are characterised by the coefficient of α4;
this can be one of 3
2
, 1
2
, −1
2
or −3
2
. The cliques have size 4, 16, 16 and 4 in the order just
mentioned. It is certainly possible that dA3ω3 = 20 = e
A3
ω3
, hence 2α4(s) = 1 and we must
136
have α4(s) = −1.
We end this chapter by considering the spin module L(ωn) for n ∈ [7, 9] with p = 2.
We may label weights in W.ωn by strings consisting of n plus and minus signs. We do
this by considering the coefficients of a weight µ ∈ W.ωn, imagining a zero coefficient
preceding the original coefficients of µ and then placing a plus sign in the ith position of
the string if the (i+1)st coefficient is larger than the ith coefficient for i ∈ [0, n−2] and we
place a plus sign in the nth position if twice the nth coefficient is larger than the (n−1)st
coefficient; otherwise there is a minus sign. This simplifies the task of determining whether
the difference between two weights is a root and to determine clusters. Two weights differ
by 2(αi + · · ·+ αj) for j < n if they are identical except that a plus and minus sign have
been interchanged in the ith and (j + 1)st positions. Recall that p = 2 so 2α(s) = 1 if
and only if α(s) = 1. Two weights differ by αn if they are identical except one has a
plus sign in the nth position of the string and the other has a minus sign. It is clear,
therefore, that if we compare two weights with one having two plus signs and the other
having two minus signs in the ith and jth positions (otherwise identical) then they differ
by 2(αi + · · ·+ αj−1 + 2αj + · · ·+ 2αn−1 + αn).
Lemma 7.8. Let G act on the irreducible modules L(ωn) for n ∈ [7, 9] with p = 2. If
n ∈ [7, 9] then (♦) is satisfied for all X ∈ I.
Proof. Recall that we have |Λ| > 2n−2 by the last paragraph in Section 7.2. If n = 9
then |Λ| > 128 and dimG = 171, whence we quickly see that we may assume that there
are two orthogonal roots, one long and one short, not in Φs. If n = 8 then |Λ| > 64 and
dimG = 136; we see that there are two orthogonal roots not in Φs unless X = C4C4.
However, if X = C4C4 then, since p = 2, the semisimple element s is a scalar multiple of
the identity, i.e., s is central.
Assume that we can take two orthogonal roots α short and β long not in the Φs. We
have 〈ωn, α〉 = 0 or ±2 and 〈ωn, β〉 = ±1. We may calculate the number of 1 × 2 and
2× 2 weight nets, but it is quicker to notice that all weights in W.ωn occur in pairs in the
weight nets; therefore |Λ| > 2n−1. If n = 9 we now find that (♦) is satisfied for all X ∈ I
137
since |Λ| > 256 > 162 = |Φ(G)|. If n = 8 then (♦) holds for all X ∈ I except X = ∅
since |Λ| > 128 = |Φ(G)|. Suppose that X = ∅ and arrange the weights into weight nets
where the horizontal difference between weights in the same net is α1 and the vertical
difference α8, say. Clearly d
∅
ω8
> 128 since the horizontal (or vertical) difference between
pairs of weights are roots and all weights appear in such pairs. However, at most one of
the four weights +++++−−+ , +++++−−− and ++−++−−+ , ++−++−−− can lie in the
eigenspace and these occur in two pairs as indicated, whence d∅ω8 > 129.
If n = 7 then in order to take two orthogonal roots as before, we need to deal with the
cases X = C5C2 and X = C4C3. However, in these cases, we see that s is a scalar multiple
of the identity, hence central. Therefore, |Λ| > 64 and dimG = 105. It remains to show
that (♦) holds for the 60 centraliser types listed in the table below; this is routine.
Centraliser type
∅, A1, A21, A
3
1, A2, A2A1, A2A
2
1, A
2
2, A3, A3A1, A3A2, A4, A4A1, A5,
C1, C1A1, C1A
2
1, C1A
3
1, C1A2, C1A2A1, C1A
2
2, C1A3, C1A3A1, C1A4, C1A5,
C21 , C
2
1A1, C
2
1A
2
1, C
2
1A2, C
2
1A2A1, C
2
1A3, C
2
1A4, C2, C2A1, C2A
2
1, C2C1, C2C1A1,
C2C1A
2
1, C2A2, C2A2A1, C2A2C1, C2A3, C2A3C1, C2A4, C
2
2 , C
2
2A1, C
2
2A2, C3, C3A1,
C3A
2
1, C3C1, C3C1A1, C3A2, C3A2C1, C3A3, C3C2, C3C2A1, C4, C4A1, C4C1
Table 7.3: Centraliser types requiring consideration for L(ω7) with n = 7
138
Chapter 8
Groups of type Dn
Throughout this chapter we shall assume that G is a simple simply connected algebraic
group of type Dn defined over an algebraically closed field K and V = L(λ) is an irre-
ducible G-module with p-restricted highest weight λ. Recall that we list modules up to
duality.
We shall prove the following result.
Theorem 8.1. Let G = Spin2n(K) act on V = L(λ). If dimV 6 dimG + 2 then
dimE = dimV with the possible exception of L(ω7) for n = 7; if instead dimV > dimG+2
then dimE < dimV with the possible exceptions of L(2ω1) for n > 4 with p 6= 2, and both
L(ω7) and L(ω8) for n = 8.
This theorem is a consequence of the lemmas which follow in later sections.
8.1 Initial survey
Consider µ 6 λ where µ =
∑n
i=1 aiωi is a dominant weight. We use Proposition 4.2
repeatedly in this section. We shall begin by obtaining conditions on the coefficients
ai in order to show that (†) is satisfied for n large enough. This will allow us later to
list modules which will require further consideration and in some cases we can use the
information on weight multiplicities in Lu¨beck’s tables to show that (†) holds.
139
Proposition 8.2. Suppose that µ =
∑n
i=1 aiωi 6 λ is a dominant weight, and at least
one of the following conditions holds:
(i) n > 6 and ak 6= 0 for some k ∈ [4, n− 2], n > 8 and a3 6= 0, and n > 11 and either
an−1 6= 0 or an 6= 0;
(ii) n > 7 and an−1, an 6= 0;
(iii) n > 5 and either ak, an−1 6= 0 or ak, an 6= 0 for some k ∈ [2, n− 2];
(iv) n > 7 and either a1, an−1 6= 0 or a1, an 6= 0;
(v) If n > 6 then m1 =
∑3
i=1 iai 6 3.
Then (†) holds.
Proof. Let us first assume that at least one coefficient ak 6= 0. Suppose that ak 6= 0 for
k ∈ [1, n− 2]. Then Ψ = 〈αi | ai = 0〉 ⊂ Φ(Ak−1Dn−k) and so
rΨ > rAk−1Dn−k = 2
k−2k
(
n
k
)
4n− 3k − 1
n(n− 1)
.
If k = 3 then rA2Dn−3 = 2(n − 2)(2n − 5) > dimG for n > 8, if k = 4 then rA3Dn−4 =
2
3
(n − 2)(n − 3)(4n − 13) > dimG for n > 6 and if k = n − 2 then rAn−3D2 = 2
n−5(n +
5)(n − 2) > dimG for n > 6. We see that for n > 6 the value of rAk−1Dn−k is at least
rA3Dn−4 for each k ∈ [4, n− 2].
Suppose that an−1 6= 0 (or equivalently an 6= 0). Then Ψ ⊂ Φ(An−1) and rΨ > rAn−1 =
2n−3 which exceeds dimG for n > 11. If both an−1 6= 0 and an 6= 0 then Ψ ⊂ Φ(An−2)
and rΨ > rAn−2 = 2
n−3(n+ 2) exceeds dimG for n > 7. Thus, (†) may not be satisfied if
n > 11 and ak 6= 0 for k ∈ [1, 2], if 8 6 n 6 10 and ak 6= 0 for k ∈ [1, 2] ∪ [n− 1, n] and if
6 6 n 6 7 and ak 6= 0 for k ∈ [1, 3] ∪ [n− 1, n].
For n > 6, if m1 =
∑3
i=1 iai > 4 then we claim that we can find {bi}
n
i=1 such that
µ′ =
∑n
j=1 bjωj < µ with b4 6= 0. It suffices to check this for the weights 4ω1, 2ω2, 2ω1+ω2,
ω1+ω3 with m1 = 4, the weights ω1+2ω2 and ω2+ω3 with m1 = 5 and 2ω3 with m1 = 6
140
since the coefficients a1, a2 and a3 of each of these weights are the smallest values that
need to be checked for any µ. Indeed all weights ν listed with m1 = 4 satisfy ν > ω4,
both weights τ with m1 = 5 listed satisfy τ > ω1 + ω4, and 2ω3 > ω2 + ω4. Hence we can
assume that m1 6 3, otherwise (†) holds.
Next suppose that a2 6= 0 and an−1 6= 0. We have Ψ ⊂ Φ(A1An−3) so that rΨ >
2n−4(n2 + 3n− 8) which exceeds dimG for n > 5. Similarly for a2 6= 0 and an 6= 0. Since
rAk−1An−k−1 > rA1An−3 for 2 6 k 6 n−2 and n > 5 we see that (†) is satisfied when ak 6= 0
and an−1 6= 0 (or an 6= 0) for such values of k and n.
Finally, suppose that a1 6= 0 and an−1 6= 0. Here Ψ ⊂ Φ(An−2) and, as we have
already seen, rΨ > dimG for n > 7 in which case (†) is satisfied. Similarly for a1 6= 0 and
an 6= 0.
We shall repeatedly use these conditions in the next three lemmas in order to obtain
a list of weights for values of n > 4 when (†) may not hold.
We shall not consider the natural module L(ω1) for n > 4 since dimL(ω1) = 2n <
n2 = dimB, i.e., the module is too small. Recall that the adjoint module is L(ω2) for
n ∈ [4,∞). The modules L(ωn−1) and L(ωn) need not be considered for n ∈ [5, 6] since
dimL(ωn−1) = dimL(ωn) = 2
n−1 < n2 for these values of n. The module L(ω7) for n = 7
has dimension 64 which is less than dimG = 91.
Lemma 8.3. Let µ =
∑n
i=1 aiωi be a dominant weight with µ 6 λ. If at least three ai are
non-zero for n > 6 then (†) is satisfied.
Proof. If n > 8 then we can have both a1 and a2 non-zero and at most one of an−1 and
an non-zero; all other coefficients can be assumed to be zero. However we cannot have
both a1 6= 0 and one of at 6= 0 for t ∈ {n− 1, n}.
If n = 7 we can again have at most one of a6 and a7 non-zero and since m1 6 3 we
have at most two ai for i ∈ [1, 3] non-zero. Thus at least one of a2 and a3 are non-zero
and one of at for t ∈ [6, 7] is non-zero, hence (†) is satisfied.
If n = 6 we may have both a5 6= 0 and a6 6= 0. Since m1 6 3 we can have at most two
ai 6= 0 for i ∈ [1, 3]. If two ai 6= 0 for i ∈ [1, 3] then we must have a1 = a2 = 1 and if we
141
have either a5 6= 0 or a6 6= 0 then (†) holds. Similarly, if only one ai 6= 0 for i ∈ [1, 3] then
we must have a1 6= 0 since (†) is satisfied if as 6= 0 and at 6= 0 for s ∈ [2, 3] and t ∈ [5, 6].
Therefore we have ai 6= 0 for i ∈ {1, 5, 6}; however ω1+ω5+ω6 > ω1+ω3 > ω4 and since
(†) is satisfied for weights with a4 6= 0 we can use Premet’s theorem for ω1 + ω5+ ω6.
We can now assume for n > 6 that at most two ai are non-zero.
Lemma 8.4. Suppose that dimV > dimG + 2. If n > 6 then (†) holds except possibly
for the modules with highest weights 2ω1 for n ∈ [6,∞) with p 6= 2, ωn−1 for n ∈ {8, 10},
ωn for n ∈ [8, 10] and ω3 for n = 6.
Proof. Assume that n > 11. We have m1 6 3 and we may assume that ak = 0 for
k ∈ [3, n]. The remaining weights to be considered are 2ω1, ω1 + ω2 and 3ω1. However,
by Premet, we need not consider either ω1+ω2 or 3ω1 since ω3 lies beneath both of these
in the partial ordering.
If 8 6 n 6 10 we again have m1 6 3 and we can assume a3 = 0 but we may also
have either an−1 = 1 or an = 1 (but not both). We cannot have either an−1 or an > 2 as
then there is a weight µ′ < µ with a′n−2 6= 0 in which case (†) is satisfied. We may not
have both as 6= 0 and at 6= 0 for s ∈ {1, 2} and t ∈ {n − 1, n}. The remaining weights
to be considered are the same three for n > 11 as well as ωn−1 and ωn. If n = 7 we may
have a3 6= 0, but since m1 6 3 the only weights other than those for 8 6 n 6 10 to be
considered are 3ω1, ω1 + ω2 and ω3.
Consider the weight ω3 for n = 7. Since ω3 > ω1 we can use Premet’s theorem to
include the contribution to |Λ| from the weights in W.ω1. We see in the case n = 7 that
rA2Dn−3 + rDn−1 = 2(n − 2)(2n − 5) + 2 just exceeds dimG. Hence (†) also holds for the
weights 3ω1 and ω1 + ω2 which lie above ω3 in the partial ordering.
If n = 6 we need to consider weights which have both a5 6= 0 and a6 6= 0; in such a
case a5 = a6 = 1 since ω4 < 2ωs for s ∈ {5, 6} and we may assume that a4 = 0. We can
have both a1 6= 0 and as 6= 0 but not both a2 6= 0 and as 6= 0 for s ∈ {5, 6}. In this case
we are forced to have as = 1 as just argued and a1 < 2 since ω2 + ωs < 2ω1 + ωs. Thus,
142
we must consider further the weights ω5 + ω6, ω1 + ω5 and ω1 + ω6 together with ω5 and
ω6 and the relevant weights satisfying m1 6 3 and aj = 0 for j ∈ [4, 6].
Consider the weight ω1 + ω6. We find for type Dn that rAn−2 = 2
n−3(n + 2) and
rAn−1 = 2
n−3 so using the fact that ω5 < ω1 + ω6 together with Premet’s theorem we
see for n = 6 that rA4 + rA5 = 72 > 66 = dimG, i.e., (†) holds. Similarly we can use
ω6 < ω1 + ω5 for the module with highest weight ω1 + ω5. Also for n = 6 we can use the
ordering ω5 + ω6 > ω3 to see that rA4 + rA2D3 = 64 + 56 = 120 which exceeds dimG.
Consider ω1+ω2 for n = 6. We have rDn−2 = 4(2n− 3) and ω1+ω2 > ω3 > ω1 so that
rD4 + rA2D3 + rD5 = 36 + 56 + 2 = 94 which exceeds dimG. By this calculation we need
not consider 3ω1 for n = 6 any further.
We note that dimL(ωn−1) = dimL(ωn) = 2
n−1 < dimG for n ∈ [4, 7] so, by duality,
we need only consider L(ωn−1) for n ∈ {8, 10} and L(ωn) for n ∈ [8, 10].
We are left to investigate the low rank cases n = 4 and 5.
Lemma 8.5. Suppose that dimV > dimG+2. If n ∈ [4, 5] then (†) holds except possibly
for the modules with highest weights 2ω1 for n ∈ [4, 5] with p 6= 2, ω3 for n = 5, both
ω1 + ωn−1 and ω1 + ωn for n ∈ [4, 5], and both 2ω4 and 2ω5 for n = 5 with p 6= 2.
Proof. If n = 5 we cannot have three or more coefficients ai 6= 0. The smallest possible
value of rΨ occurs for A
2
1, however, rA21 = 216 which certainly exceeds dimG = 45.
Therefore, assume that at most two ai are non-zero. We cannot have both as 6= 0
and at 6= 0 for s ∈ {2, 3} and t ∈ {4, 5}. Suppose that both a1 6= 0 and at 6= 0 for
t ∈ {4, 5}. Since ω2 + ωt < 2ω1 + ωt we must have a1 = 1. Using Premet’s theorem, since
ω1 + ω3 < ω1 + 2ωt and rA3
1
= 102 we see that (†) is satisfied and the only weights left to
be considered are ω1 + ω4 and ω1 + ω5.
Suppose that only one of either a4 6= 0 or a5 6= 0. Since ω3+ω4 < 3ω4 and ω3+ω5 < 3ω5
and rA2A1 = 64 we are left to consider the weights 2ω4 and 2ω5.
Similarly, we cannot have both a4 6= 0 and a5 6= 0 since ω2 < ω4+ω5 and rA1A3+rA3 =
18 + 28 = 46 > 45 = dimG. We can assume from now on that a4 = a5 = 0. We can
have neither ω2 + ω3 nor ω1 + ω3 since rA3
1
= 102. Similarly we cannot have ω1 + ω2 since
143
ω3 < ω1 + ω2 and rA3 + rA2A21 = 28 + 30 = 58. Therefore we cannot have any two of ai
for i ∈ [1, 3] non-zero. We can assume that a1 6 2 since 3ω1 > ω1 + ω2. The following
ordering of weights 2ω3 > 2ω2 > ω1 + ω3 means we may assume that a2 6 1 and a3 6 1
since rA3
1
= 102. Thus the weights remaining to be considered are 2ω1 and ω3.
Consider the case n = 4. If at least three ai 6= 0 we see that the smallest value of rΨ
occurs, of course, for Ψ = A1 in which case rA1 = 44; this exceeds dimG = 28 and (†) is
satisfied. We can assume that at most two ai are non-zero. First suppose that a2 6= 0.
Since 2ω2 > ω1 + ω3 + ω4 we can use Premet’s theorem to conclude that (†) holds and
therefore we may assume that a2 < 2. We do not need to consider ω1 + ω2 further since
ω1 + ω2 > ω3 + ω4 and we calculate rA2
1
= 20 and rA2 = 12. By symmetry we see that
if a2 = 1 then a1 = a3 = a4 = 0. Now suppose that a2 = 0. As we have just seen, we
cannot have 3ω1 since 3ω1 > ω1+ω2. Similarly we cannot have the weight 2ω1+ω3 since
2ω1+ω3 > ω2+ω3. Thus the weights which require further consideration are 2ω1, ω1+ω3
and ω1 + ω4.
The following table displays (up to duality) the highest weights of irreducible p-
restricted G-modules which require further consideration.
λ n Lemma
2ω1 [4,∞) (p 6= 2) -
ωn−1 10 8.7 (†)
8 8.7 (♦)a
ωn 10 8.7 (†)
[8, 9] 8.7 (♦)a
ω3 6 8.6 (†)
5 8.6 (♦)
ω1 + ωn−1 5 8.6 (†)
4 8.6 (♦)
ω1 + ωn 5 8.6 (†)
4 8.6 (♦)
2ω4 5 (p 6= 2) 8.6 (†)
2ω5 5 (p 6= 2) 8.6 (†)
a Except for X = A7 when n = 8
Table 8.1: Possible weights in type Dn for n > 4
144
8.2 Weight string analysis
We shall begin with the modules which require the consideration of few, if any, centraliser
types.
Lemma 8.6. Let G act on the irreducible module V where we take V to be one of L(ω1+
ωn−1) or L(ω1 + ωn) for n ∈ [4, 5], L(2ω4) or L(2ω5) for n = 5 with p 6= 2, or L(ω3) for
n ∈ [5, 6]. Then (†) holds for L(2ω4), L(2ω5), L(ω1 + ω4) and L(ω1 + ω5) for n = 5, and
L(ω3) for n = 6, otherwise (♦) holds for each X ∈ I.
Proof. We perform the usual calculations for L(2ω5) with n = 5 and display them in
Figure 8.1. Since dimD5(K) = 45 we see that (†) holds. Similarly for L(2ω4) with n = 5.
i ω |W.ω| mω
p 6= 2
3 2ω5 16 1
2 ω3 80 1
1 ω1 10 3
Weight No. of l
strings strings p 6= 2
µ3 8
µ3 µ3 4 4
µ2 20
µ2 µ2 24 24
µ2 µ1 µ2 6 12
µ1 µ1 2 6
Lower bound on |Λ| 46
Figure 8.1: (λ, n) = (2ω5, 5)
The use of weight multiplicities if n = 6 for the module L(ω3) in Figure 8.2 shows that
(†) holds. If n = 5 then Figure 8.3 shows that (♦) holds for all X ∈ I except for X = ∅
with p = 2.
i ω |W.ω| mω
p 6= 2 p = 2
2 ω3 160 1 1
1 ω1 12 5 4
Weight No. of l
strings strings p 6= 2 p = 2
µ2 48
µ2 µ2 48 48 48
µ2 µ1 µ2 8 16 16
µ1 µ1 2 10 8
Lower bound on |Λ| 74 72
Figure 8.2: (λ, n) = (ω3, 6)
Thus we can take two orthogonal roots not in Φs; we consider pairs of such roots not of
the form ǫi+ ǫj and ǫi− ǫj. We shall calculate the number of weight nets. Set x = 〈ω3, α〉
and y = 〈ω3, β〉. We note that there are the same number of pairs of orthogonal roots with
(x, y) = (a, b) and (b, a) provided a 6= b. There are 12 pairs of orthogonal roots α, β such
145
i ω |W.ω| mω
p 6= 2 p = 2
2 ω3 80 1 1
1 ω1 10 4 2
Weight No. of l
strings strings p 6= 2 p = 2
µ2 20
µ2 µ2 24 24 24
µ2 µ1 µ2 6 12 12
µ1 µ1 2 8 4
Lower bound on |Λ| 44 40
Figure 8.3: (λ, n) = (ω3, 5)
that (x, y) = (2, 1), 12 pairs such that (x, y) = (2, 0), 48 pairs such that (x, y) = (1, 1)
and 24 pairs such that (x, y) = (1, 0). Thus, for a particular pair α, β there are 80.12
480
= 2
weights µ ∈W.ω3 such that (x, y) = (2, 1), i.e., there are two 3× 2 weight nets. Similarly,
we find that there are two 2× 3 weight nets, two 3× 1 and two 1× 3 weight nets, eight
2 × 2 weight nets, and four 2 × 1 and four 1 × 2 weight nets. Note that weight strings
of length 3 occurring in the weight nets are of the form µ2 µ1 µ2 (taking the notation of
Figure 8.3).
There are 96 pairs of orthogonal roots with 〈ω1, α〉 = 〈ω1, β〉 = 0. So, for a given pair
of roots α and β, there are 10.96
480
= 2 weights µ ∈ W.ω1 orthogonal to both α and β. Thus
the two 3× 1 and two 1× 3 weight nets above actually form two weight nets which have
a plus-shape. The same weight in W.ω1 occurs in one of the 1 × 3 and one of the 3 × 1
weight nets.
Thus |Λ| > 56 or 52 according as p 6= 2 or p = 2 whence (♦) always holds.
Next we consider L(ω1 + ωn) for n ∈ [4, 5]. If n = 5 then Figure 8.4 shows that (†)
holds, whereas if n = 4 further work is required to show that (♦) is satisfied forX = ∅ and
A1 for all p and D2 for p = 2 as is evident from Figure 8.5. We note that the subsystem
A21 does not occur here.
i ω |W.ω| mω
p 6= 5 p = 5
2 ω1 + ω5 80 1 1
1 ω4 16 4 3
Weight No. of l
strings strings p 6= 5 p = 5
µ2 24
µ2 µ2 20 20 20
µ2 µ1 µ2 8 16 16
µ1 µ1 4 16 12
Lower bound on |Λ| 52 48
Figure 8.4: (λ, n) = (ω1 + ω5, 5)
If X = ∅ then the following cliques and their negatives show that d∅ω1+ω4 > 32 or 26
according as p 6= 2 or p = 2. Therefore (♦) is satisfied since e∅ω1+ω4 = 24. Note that the
146
i ω |W.ω| mω
p 6= 2 p = 2
2 ω1 + ω4 32 1 1
1 ω3 8 3 2
Weight No. of l
strings strings p 6= 2 p = 2
µ2 8
µ2 µ2 8 8 8
µ2 µ1 µ2 4 8 8
µ1 µ1 2 6 4
Lower bound on |Λ| 22 20
Figure 8.5: (λ, n) = (ω1 + ω4, 4)
first cluster below consists of weights in W.ω3.
1
2
1 1 1
2
3
2
2 1 3
2
3
2
2 1 1
2
1
2
0 1 1
2
3
2
1 1 1
2
1
2
1 0 1
2
1
2
2 1 3
2
1
2
2 1 1
2
- 1
2
0 1 1
2
3
2
1 0 1
2
1
2
0 0 1
2
1
2
1 1 3
2
1
2
1 1 - 1
2
- 1
2
-1 0 1
2
- 1
2
0 0 1
2
In the table below we handle the case X = A1 with simple root α1. There are six
cliques involving weights in W.ω3 as can be seen in the right-hand table in Figure 8.5.
The table below shows the arrangement of some clusters into cliques; taking them and
the corresponding negative cliques yields dA1ω1+ω4 > 28 or 24 according as p 6= 2 or p = 2
which in either case exceeds eA1ω1+ω4 = 22.
X = A1
Clique Cluster size l Clique Cluster size l
· 2 1 3
2
2 2 · 1 1 1
2
5/4 10/8
· 2 1 1
2
2 · 1 0 1
2
5/4
· 0 1 1
2
2 2 · 0 0 1
2
6/4
· 0 1 - 1
2
2
We are left with X = D2 with simple roots α3 and α4. There are two cliques consisting
of six weights and containing two weights in W.ω3. Using the clusters given in the table
below together with their negatives, we see that dD2ω1+ω4 > 28 or 24 according as p 6= 2 or
p = 2. Since eD2ω1+ω4 = 20 we have that (♦) is satisfied.
X = D2
Clique Cluster size l Clique Cluster size l Clique Cluster size l
1
2
1 · · 10/8 10/8 3
2
2 1 · 2 2 1
2
2 1 · 2 2
1
2
0 · · 10/8 3
2
1 · 1
2
2 - 1
2
1 · 1
2
2
Similarly for L(ω1 + ωn−1) with n ∈ [4, 5].
Our methods do not suffice for L(2ω1). Using weight multiplicities the best we can
show is that |Λ| > 4(n− 1). This is not conducive to further progress.
147
It remains to consider the spin modules L(ωn) for n ∈ [8, 10] and L(ωn−1) for n ∈
{8, 10}. We recall that Φ(Dn) = {±ǫi ± ǫj | 1 6 i < j 6 n}. Let αi = ǫi − ǫi+1 for
i ∈ [1, n− 2], αn−1 = ǫn−1 − ǫn and αn = ǫn−1 + ǫn be the simple roots.
Lemma 8.7. Let G act on the irreducible modules L(ωn−1) for n ∈ {8, 10} and L(ωn) for
n ∈ [8, 10]. Then (†) holds for both L(ω9) and L(ω10) with n = 10, (♦) is satisfied for all
X ∈ I for L(ω9) with n = 9, and (♦) is satisfied for all X ∈ I \ {A7} for both L(ω7) and
L(ω8) with n = 8.
Proof. Recall that dimL(ωn) = 2
n−1 and rAn−1 = 2
n−3. We shall explain how we may
write each weight in this module as a string of n plus and minus signs. Consider the
coefficients bi of a weight µ written as a linear combination of simple roots µ =
∑n
i=1 biαi
and setting b0 = 0 we imagine a zero coefficient preceding b1. If bi > bi−1 for 1 6 i 6 n−2
and i = n there is a plus sign in position i; otherwise there is a minus sign. There is a plus
sign in position n − 1 if bn−2 < bn−1 + bn; otherwise there is a minus sign. Suppose that
two weights when written as a string of plus and minus signs are identical apart from the
ith and jth positions where the first weight has a plus sign in position i and minus sign
in position j and vice-versa for the second weight. Then the difference between them is
the root αi + · · ·αj−1. Consider two weights identical except one has two plus signs and
the other has two minus signs in the (n− 1)st and nth positions. The difference between
these two weights is αn. Therefore, the difference between two weights that are identical
other than one has two plus signs and the other has two minus signs in the ith and jth
positions is the root αi + · · ·+ αj−1 + 2αj + · · ·+ 2αn−2 + αn−1 + αn. We note that there
are always an even number of minus signs in the string of plus and minus signs for a given
weight. There is one weight consisting of n plus signs (and no minus signs),
(
n
2
)
weights
with n− 2 plus signs,
(
n
4
)
weights with n− 4 plus signs, etc.
First assume that n = 10 so that |Λ| > 128. There is a dichotomy here: either
X = A9 or we can take the two orthogonal roots ǫ1 + ǫ10 = α1 + · · · + α8 + α10 and
ǫ1 − ǫ10 = α1 + · · ·+ α9 not in Φs.
In the former case we provide the clusters that occur in the table below. It is clear
148
that dA9ω10 > 210 > 190 = dimG.
X = A9
Cluster Cluster size
+ + ++++++++ : 1
+ +++++++−− : 45
+ + ++++−−−− : 210
+ + ++−−−−−− : 210
+ +−−−−−−−− : 45
−−−−−−−−−− : 1
In the latter case, taking the two orthogonal roots outside Φs, each weight occurs in
a pair differing by one of these two roots; given below are the four types of pairs. Note
that weights in a pair have identical signs in the second to eighth positions.
+ • • • • • • •++ + • • • • • • • − − + • • • • • • •+− + • • • • • • • −+
− • • • • • • •+− − • • • • • • • −+ − • • • • • • •++ − • • • • • • • − −
Thus by the adjacency principle we find that |Λ| > 256 > 190 = dimG so (†) holds.
Assume that n = 9 so that |Λ| > 64. We can always take the two orthogonal roots
ǫ1 + ǫ9 and ǫ1 − ǫ9 outside Φs unless X = A8. However, as can be seen from the table
below, for X = A8 if the cluster + + + + + − − − − lies in or not in the eigenspace, (♦) is
satisfied.
X = A8
Cluster Cluster size
+ + +++++++ : 1
+ ++++++−− : 36
+ + +++−−−− : 126
+ + +−−−−−− : 84
+−−−−−−−− : 9
Now by taking the two orthogonal roots not in Φs we have |Λ| >
1
2
dimL(ω9) = 128 as
before. Thus (♦) holds for all X ∈ I satisfying |Φ(X)| > 144− 128 = 16. The remaining
centraliser types are detailed in Table 8.2 and can be dealt with easily.
Assume that n = 8 so |Λ| > 32. As above there are two cases: either X = A7
or the orthogonal roots ǫ1 + ǫ8 and ǫ1 − ǫ8 are not in Φs. In the latter case we have
|Λ| > 1
2
dimL(ω8) = 64 so (♦) is satisfied unless |Φ(X)| > 112 − 64 = 48 and the
remaining centraliser types are given in Table 8.2. These are reasonably straightforward
149
to go through. In the former case we arrange the 128 weights in W.ω8 into five clusters
in the table below.
X = A7
Cluster Cluster size
+ + ++++++ : 1
+ +++++−− : 28
+ + ++−−−− : 70
+ +−−−−−− : 28
−−−−−−−− : 1
It is possible that dA7ω8 = e
A7
ω8
= 56 by assuming that the second and fourth clusters in
the table below lie in Λ. Thus we cannot conclude that (♦) holds for X = A7.
n Centraliser types
9 ∅, A1, A21, A
3
1, A
4
1; A2, A2A1, A2A
2
1, A2A
3
1, A
2
2, A
2
2A1; A3, A3A1, A3A
2
1;
D2, D2A1, D2A
2
1, D2A
3
1, D2A2, D2A2A1, D2A2A
2
1, D2A
2
2, D2A3, D
2
2,
D22A1, D
2
2A
2
1, D
2
2A2, D
2
2A2A1; D3, D3A1, D3A
2
1, D3D2
8 ∅, A1, A21, A
3
1, A
4
1; A2, A2A1, A2A
2
1, A
2
2, A
2
2A1; A3, A3A1, A3A
2
1, A3A2, A
2
3;
A4, A4A1, A4A2; A5, A5A1; A6; D2, D2A1, D2A
2
1, D2A
3
1, D2A2, D2A2A1,
D2A
2
2, D2A3, D2A4, D2A5, D
2
2, D
2
2A1, D
2
2A
2
1, D
2
2A2, D
2
2A3; D3, D3A1, D3A
2
1,
D3A2, D3A2A1, D3D2, D3D2A1, D3D2A2, D3A3, D3A4, D
2
3, D
2
3A1; D4, D4A1,
D4A
2
1, D4A2, D4A3, D4D2, D4D2A1, D4D3, D
2
4; D5, D5A1, D5A2, D5D2
Table 8.2: Centraliser types requiring consideration for L(ωn) with n ∈ [8, 9]
150
Chapter 9
Groups of exceptional type
In this chapter we consider the groups of exceptional type acting on p-restricted irreducible
modules. We shall use Proposition 4.2 repeatedly in order to show that (†) holds for nearly
all modules V with dimV > dimG + 2. If there are roots of different lengths we shall
use the standard notation A˜j to denote the root subsystem Aj for j ∈ [1, n] consisting
of short roots. We remark that if G is a simple algebraic group of type E8, F4 or G2
then there is only one group in the isogeny class, i.e., the adjoint and simply connected
group are the same. However, for all other simple algebraic groups there is a need to
distinguish between the simply connected group and other isogenous groups such as the
adjoint group.
9.1 Types E6, E7 and E8
Lemma 9.1. Consider the simple simply connected algebraic group G of type En for
n ∈ {6, 7, 8} defined over an algebraically closed field K acting on an irreducible G-module
with p-restricted highest weight λ. If dimV > dimG+ 2 then (†) holds.
Proof. Consider a dominant weight µ =
∑n
i=1 aiωi with µ 6 λ. Suppose that ak 6= 0 for
some k ∈ [1, n]. Then Ψ = 〈αi | ai = 0〉 ⊂ Ψk = 〈αi | i 6= k〉 and rΨ > rΨk for each k.
The possible Ψk and values of rΨk for each k and n are given in the table below.
151
Type E8 Type E7 Type E6
k Ψk rΨk Ψk rΨk Ψk rΨk
1 D7 702 D6 33 D5 6
2 A7 6624 A6 192 A5 21
3 A1A6 28224 A1A5 752 A1A4 75
4 A2A1A4 213696 A2A1A3 4240 A2A1A2 290
5 A4A3 104832 A4A2 1600 A4A1 75
6 D5A2 122220 D5A1 252 D5 6
7 E6A1 2324 E6 12
8 E7 57
Assume that n = 8. Clearly rΨ exceeds dimG = 248 for k 6= 8 so (†) holds. Thus
we need only consider weights of the form µ = a8ω8. We quickly see that only a8 = 1 is
possible by using Premet’s theorem since ω7 < 2ω8. The module of highest weight ω8 is
the adjoint module.
Now assume that n = 7 in which case dimG = 133. We see that if ak 6= 0 for k ∈ [2, 6]
then (†) holds and we need only consider modules of the form µ = a1ω1 + a7ω7. However
we have 2ω1 > ω3, 2ω7 > ω6 and ω1+ω7 > ω2. Therefore, the only weights remaining are
ω1 and ω7 which correspond to the adjoint module and a 56-dimensional module L(ω7);
recalling that for type E7 the dimension of a Borel subgroup is 70 finishes the analysis for
type E7.
Finally, if n = 6 then dimG = 78 and we can only conclude from the table above that
we may assume that a4 = 0 as otherwise (†) holds. Note that there is a non-trivial graph
automorphism for E6 so we need only consider modules up to duality. Suppose only one
of the ak 6= 0 for k 6= 4. If a1 > 3 then (†) holds since 3ω1 > ω1 + ω2 > ω3 and the same
is true if a3 > 2 since 2ω3 > ω4. By duality we may assume that a6 6 2 and a5 6 1.
We are done if a2 > 2 since 2ω2 > ω4. It can be easily checked that 2ω1 > ω3 > ω6 and
from the table we see that rA1A4 = 75 and rD5 = 6. This shows that (†) holds for the
irreducible modules with highest weights ω3 and 2ω1. We are left with the irreducible
modules L(ω2) and (up to duality) L(ω1); however the former is the adjoint module and
we do not consider the latter since dimL(ω1) = 27 < dimB = 42.
Suppose now we have at least two ak 6= 0. The subsystem of E6 of rank 4 with largest
152
Weyl group and the most roots is D4. Thus in order to obtain the smallest value of rΨ
when Ψ has rank at most 4 we must take Ψ = D4. Since rD4 = 90 exceeds dimG we may
assume that only one ak is non-zero; we have just dealt with this situation.
9.2 Type F4
Recall that we need only consider short roots when there are two different root lengths.
Lemma 9.2. Consider the simple algebraic group G = F4(K) acting on an irreducible
G-module with p-restricted highest weight λ. If dimV > dimG+ 2 then (†) holds.
Proof. We need not consider the module L(ω4) which has dimension 25 or 26 according
as p = 3 or p 6= 3 nor the adjoint module L(ω1). Let µ =
∑4
i=1 aiωi 6 λ be a dominant
weight. We assume first that p 6= 2 in order to use Premet’s theorem to obtain conditions
on the coefficients of µ. Suppose at least two ak 6= 0. Then rankΨ 6 2 and the possible
subsystems of rank 2 in F4 are A˜2, A2, A1A˜1 and B2; the smallest rΨ occurs for Ψ = B2
where rB2 =
1
2
1152
23
(24−4)
24
= 60 which exceeds dimG = 52. So we are left to consider
modules with one non-zero coefficient. We do not need to consider further any weight
of the form a1ω1 since 2ω1 > ω1 + ω4 and as mentioned above we cannot have a1 = 1.
Similarly if a3 6= 0 then (†) holds by Premet’s theorem since ω3 > ω1 > ω4 and we find
rA2A˜1 = 44, rC3 = 6 and rB3 = 9. If a4 6= 0 and a4 > 1 then (†) holds since 2ω4 > ω3.
Again, we cannot have a2 6= 0 since ω2 > ω1 + ω4. Thus the result holds for p 6= 2.
Now assume p = 2. We need to take care in this case as weight strings may not be
saturated, hence we must rely on the information in [18] and [9]. However neither of
these references provide information about ω2 + ω3 for p = 2 or weights with at least
three coefficients non-zero. We shall use the fact that, for a given weight µ and short root
α ∈ Φ, if 〈µ, α〉 = ±2m for m > 0 then the adjacency principle applies because α(s) = 1
if and only if 2mα(s) = 1 for any s ∈ Gss \ Z.
First suppose that only one ak 6= 0. We need only consider the module L(ω2) for
p = 2 by the first sentence of the lemma and since there is a graph automorphism of the
153
group for p = 2. We see from Figure 9.1 that (†) holds for p = 2. We have included
all characteristics to highlight the situation for weight strings when the conclusion of
Premet’s theorem does not hold; we can ignore all µi with i ∈ {1, 3, 5} for p = 2 as these
weights occur with multiplicity zero.
i ω |W.ω| mω
p 6= 2, 3 p = 3 p = 2
6 ω2 96 1 1 1
5 ω1 + ω4 144 1 1 0
4 2ω4 24 3 3 2
3 ω3 96 4 4 0
2 ω1 24 10 9 4
1 ω4 24 13 12 0
0 0 1 26 22 6
Weight No. of l
strings strings p 6= 2, 3 p = 3 p = 2
µ6 24
µ6 µ4 µ6 24 48 48 24
µ6 µ3 µ2 µ3 µ6 12 96 96 24
µ5 µ5 24 24 24 0
µ5 µ4 µ5 6 12 12 0
µ5 µ3 µ3 µ5 24 120 120 0
µ5 µ2 µ1 µ2 µ5 6 90 84 24
µ4 µ3 µ4 8 32 32 16
µ4 µ1 µ0 µ1 µ4 1 26 24 4
µ3 µ1 µ1 µ3 8 136 128 0
Lower bound on |Λ| 584 568 92
Figure 9.1: λ = ω2
Suppose that two ak 6= 0. Suppose that µ = ω3 + ω4 in which case we have 〈µ, α〉 = 1
for four short roots (0010, 0001, 0110, and 1110), 〈µ, α〉 = 2 for three short roots (0011,
0111 and 1111) and 〈µ, α〉 = 4 for one short root (1231). Thus, for a given α ∈ ΦS,
there are 192.8
24
= 64 weights µ ∈ W (Φ) that appear in weight strings along with µ− 2mα
for some m ∈ [0, 2], so there is a contribution of at least 64 to |Λ| and (†) is satisfied.
Using Lu¨beck’s tables [18] we can use this contribution also for the weights ω1 + ω3 and
ω2 + ω4 since ω3 + ω4 appears with non-zero multiplicity lower in the partial ordering of
both of these weights. We calculate in an exactly similar way that ω2 + ω3, ω1 + ω2 and
2ω3 contribute at least 84, 48 and 36 to |Λ|. Since by [18] we know that 2ω3 occurs in
L(ω1 + ω2) with a non-zero multiplicity, (†) holds. From [18] we see that ω3, ω1 and ω4
occur in L(ω1+ ω4) with non-zero multiplicity and that the same is true for ω1 and ω4 in
L(ω3). Therefore we can use the same calculation above for L(ω3) with p 6= 2 to conclude
that (†) holds for L(ω1 + ω4).
Suppose that three ak are non-zero. If µ = ω1 + ω3 + ω4 then the number of short
roots α ∈ Φ(F4) such that 〈µ, α〉 = 1, 2 or 4 is three (0010, 0001 and 0110), two (0011
and 0111) or one (1111), respectively. Thus, for a given α ∈ ΦS, there are
576.6
24
= 144
weights µ ∈ W (Φ) that appear in weight strings along with µ− 2mα for some m ∈ [0, 2]
154
implying that there is a contribution of at least 144 to |Λ| by the adjacency principle.
The calculations for ω1 + ω2 + ω3, ω1 + ω2 + ω4 and ω2 + ω3 + ω4 and ω1 + ω2 + ω3 + ω4
are analogous, and also show that (†) holds.
We remark that we could have used Zalesski’s theorem for the module L(ω3+ω4) with
p = 2 since it has zero support on the long roots whence it is tensor indecomposable. The
same is not true though for L(ω1+ω2) with p = 2 even though it has zero support on the
short roots; this is one of the exceptions for type F4 in Theorem 2.7.
9.3 Type G2
In this section we consider the action of G2(K) on irreducible p-restricted modules.
Lemma 9.3. Consider the simple algebraic group G = G2(K) acting on an irreducible
G-module with p-restricted highest weight λ. If dimV > dimG + 2 then (†) holds except
for the modules L(2ω1) for p 6= 2 and L(2ω2) for p = 3 where (♦) is satisfied.
Proof. The conditions in the statement of the lemma mean that we are not considering the
modules L(ω1) and L(ω2) since dimG = 14. We calculate r∅ = 6, rA˜1 = 2 and rA1 = 3.
We shall assume for the moment that p 6= 2, 3. Let µ = a1ω1 + a2ω2 6 λ be a dominant
weight. If a2 = 0 then we may assume that a1 = 2 since 3ω1 > ω1 + ω2 > 2ω1 > ω2 > ω1
in which case |Λ| > 17. Similarly, if a1 = 0 then (†) holds by Premet’s theorem if a2 > 2
since 2ω2 > 3ω1.
Now suppose that both a1 6= 0 and a2 6= 0. Since
ω1 + 2ω2 > 2ω1 + ω2 > 2ω2 > ω1 + ω2 > ω2
we see that (†) holds if a1+a2 > 2 using the values of rΨ given above and Premet’s theorem.
We shall consider the module L(ω1 + ω2) later (where we use weight multiplicities) since
the partial ordering ω1 + ω2 > 2ω1 > ω2 > ω1 only shows that |Λ| > 6 + 3 + 2 + 3 = 14.
Assume that p = 2; the only weight we need to consider is ω1+ω2 since the other two
2-restricted modules are too small as we have mentioned.
155
Assume that p = 3. Then the weights ω1 + ω2, 2ω1 and 2ω2 require further analysis.
We see from [18] that the same ordering of weights used for p > 3 shows that (†) holds
for ω1 + 2ω2 and 2ω1 + ω2. The remaining 3-restricted weight 2ω1 + 2ω2 is only listed by
Lu¨beck on his website for p = 5 and 7 as otherwise the dimension of the module exceeds
500. However the required information is contained in the paper of Gilkey and Seitz in [9,
p.413] from which we see that for p > 2 all dominant weights µ with µ 6 2ω1+2ω2 occur
with non-zero multiplicity. Thus (†) holds as 2ω1 + 2ω2 > ω1 + 2ω2. We remark that for
p = 3 the multiplicities of dominant weights below 2ω1 + 2ω2 in the partial ordering are
the same as those obtained using Freudenthal’s formula [13, p.124] for characteristic 0.
It remains to consider the modules L(2ω1) for p 6= 2, L(ω1 + ω2) for all p and L(2ω2)
for p = 3.
We begin with L(2ω1) and detail the calculations involving weight strings in Figure
9.2.
i ω |W.ω| mω
p 6= 2, 7 p = 7
3 2ω1 6 1 1
2 ω2 6 1 1
1 ω1 6 2 2
0 0 1 3 2
Weight No. of l
strings strings p 6= 2, 7 p = 7
µ3 µ2 µ3 2 2 2
µ3 µ1 µ0 µ1 µ3 1 4 4
µ2 µ1 µ1 µ2 2 6 6
Lower bound on |Λ| 12 12
Figure 9.2: λ = 2ω1
We see that (♦) is satisfied for all centraliser types except when X = ∅. It suffices
to consider only the case p = 7. We assume in turn that the weight 0 is either in or not
in the eigenspace. If 0 is in the eigenspace then all weights in W.ω1 or W.ω2 lie in Λ, so
d∅2ω1 > 6 + 2.6 = 18 > 12 = e
∅
2ω1
. If 0 is not in the eigenspace then the two cliques shown
below together with their negative counterparts show that d∅2ω1 > 2+2.4+2.2 = 14. The
weights in W.ω1 are italicised since they have multiplicity 2.
4 2 2 1
3 2 1 1
3 1 1 0
Figure 9.3 shows that (†) is satisfied for L(ω1 + ω2) for all p.
Consider the module L(2ω2). We see from Figure 9.4 that if p = 3 then (♦) holds
except when X = ∅. (The case p 6= 2, 3 is included in the figure to highlight that the
156
i ω |W.ω| mω
p 6= 3, 7 p = 3 p = 7
4 ω1 + ω2 12 1 1 1
3 2ω1 6 2 2 1
2 ω2 6 2 1 1
1 ω1 6 4 3 2
0 0 1 4 1 2
Weight No. of l
strings strings p 6= 3, 7 p = 3 p = 7
µ4 µ4 2 2 2 2
µ4 µ3 µ2 µ3 µ4 2 8 6 4
µ4 µ2 µ1 µ1 µ2 µ4 2 14 10 8
µ3 µ1 µ0 µ1 µ3 1 8 5 4
Lower bound on |Λ| 32 23 18
Figure 9.3: λ = ω1 + ω2
conclusion of Premet’s theorem does not hold when p = 3.)
i ω |W.ω| mω
p 6= 2, 3 p = 3
6 2ω2 6 1 1
5 3ω1 6 1 1
4 ω1 + ω2 12 1 0
3 2ω1 6 2 0
2 ω2 6 3 2
1 ω1 6 3 0
0 0 1 5 3
Weight No. of l
strings strings p 6= 2, 3 p = 3
µ6 2
µ6 µ4 µ3 µ2 µ3 µ4 µ6 2 10 4
µ5 µ4 µ4 µ5 2 4 2
µ4 µ2 µ1 µ1 µ2 µ4 2 14 4
µ5 µ3 µ1 µ0 µ1 µ3 µ5 1 8 2
Lower bound on |Λ| 36 12
Figure 9.4: λ = 2ω2
If X = ∅ we can arrange the weights into clusters using the fact that α(s) = 1 if and
only if 3α(s) = 1 for any s ∈ Gss \ Z. We consider separately the cases when the weight
0 lies and does not lie in the eigenspace. We shall italicise weights in W.ω2 since these
occur with multiplicity 2. If 0 is in the eigenspace then the weights 3 2, 3 1, 0 1, 3 0 and
their negatives lie in Λ, so d∅2ω2 > 14 > 12 = e
∅
2ω2
. If 0 is not in the eigenspace then the
three cliques below show that d∅2ω2 > 3 + 2 + 4 + 4 = 13, as required.
6 4 3 2 0 -1
6 3 3 1 -3 -1
3 3 0 1 -3 -2
157
Chapter 10
Twisted modules
In this chapter we consider simple simply connected algebraic groups defined over an
algebraically closed field K of characteristic p > 0 acting on tensor products with twists
of irreducible p-restricted modules; we shall call these twisted modules. Recall that we
need only consider irreducible modules V = U ⊗W (q) where both dimU and dimW are
less than dimG; this quickly follows from Proposition 4.3. It is also clear that we need
not consider modules that are tensor products with twists of three or more p-restricted
irreducible modules. For classical groups the p-restricted modules of lowest dimension
are the natural modules and the dimension of the tensor product of two of these for any
simple algebraic group exceeds dimG.
We shall prove the following result.
Theorem 10.1. Let G be a simple simply connected algebraic group acting on an irre-
ducible module V = L(µ)⊗ L(ν)(q) where µ and ν are p-restricted and q is a power of p,
and dimV > dimG + 2. Then dimE < dimV with the possible exceptions of the action
of a group of type Bn, Cn or Dn on L(ω1)⊗ L(ω1)
(q).
This theorem is a consequence of the lemmas which follow in later sections.
158
10.1 Initial survey
We shall use Proposition 4.3 to show that (†) holds for all but a small list of irreducible
modules. If V = U ⊗W (q) is such a module then we investigate weight strings for both U
and W and then combine the weight strings together to form weight nets; in this way we
can find a lower bound for the codimension of any eigenspace of V . Consider the G-module
L(λ)(q) where q = pr with r > 0 (note that we used slightly different notation in Theorem
2.3). The weights of this module are just qµ where µ ∈ Π(λ). In some sense, the twist by
q does not affect the adjacency principle since the equation qµ(s) = q(µ+ α)(s) for some
s ∈ Gss \ Z(G) is equivalent to qα(s) = α(s) = 1. In particular in the tensor product of
two p-restricted modules the twist by q may be applied to either module without affecting
our analysis. We choose, therefore, without loss of generality to apply the twist to the
p-restricted irreducible module of larger dimension. We note that the action of a group
on U ⊗ V (q) is not necessarily the same as that of the same group on U∗ ⊗ V (q); there is
an example of this in Section 3.2.
Proposition 10.2. Let G be a simple simply connected algebraic group and let V =
U ⊗W (q) where U and W are irreducible p-restricted G-modules. Then (†) holds except
possibly for the irreducible modules given in Table 10.1.
Proof. Case I: An. Suppose that G is of type An for n > 1, so dimG = n(n+2). If n = 1
we need to consider the modules L(ω1)⊗L(2ω1)
(q) and L(2ω1)⊗L(2ω1)
(q). If n > 2 then
we need to consider the p-restricted modules L(ω1) of dimension n+1, L(ω2) of dimension
1
2
n(n+1), L(2ω1) of dimension
1
2
(n+1)(n+2), the adjoint module L(ω1+ωn) of dimension
n(n + 2) if p ∤ n + 1 and n(n + 2) − 1 otherwise and L(ω3) for n ∈ [5, 7] of dimension
1
6
n(n2− 1). We calculate lower bounds (in the same order just used) for the codimension
of a given eigenspace for these five modules: these are rAn−1 = 1, rA1An−2 = n − 1,
rAn−1 + rA1An−2 = n (since 2ω1 > ω2), rAn−2 = 2n− 1 and rA2An−3 =
1
2
(n− 1)(n− 2).
We see that (†) holds for L(ω1) ⊗ L(ω1 + ωn)
(q) since rAn−2 dimL(ω1) > dimG for
n > 2. Since L(ω1+ωn) is the adjoint module and |Λ| > 1 for all modules to be considered
except L(ω1) (which we have just dealt with), we see that (†) holds for any twisted module
159
involving L(ω1+ωn). Also (†) holds for L(ω1)⊗L(ω3)
(q) since rA2An−3 dimL(ω1) > dimG
for n ∈ [5, 7] and L(ω2)⊗L(ω3)
(q) since rA1An−2 dimL(ω3) > dimG for n ∈ [5, 7], whence it
also holds for L(ω3)⊗L(ω3)
(q). The modules L(2ω1)⊗L(2ω1)
(q) and L(2ω1)⊗L(ω3)
(q) both
satisfy (†) since (rAn−1 +rA1An−2) dimL(2ω1) > dimG for n > 2 and rA2An−3 dimL(2ω1) >
dimG for n ∈ [5, 7]. Similarly (†) holds for the modules L(ω2)⊗L(2ω1)
(q) for n > 3 since
rA1An−2 dimL(2ω1) > dimG and L(ω2) ⊗ L(ω2)
(q) for n > 4 since rA1An−2 dimL(ω2) >
dimG.
We shall need to investigate further U ⊗ V (q) where U ∈ {L(ω1), L(ωn)} and V ∈
{L(ω2), L(ωn−1)} for n > 3, U⊗V
(q) where U ∈ {L(ω1), L(ωn)} and V ∈ {L(2ω1), L(2ωn)}
for n > 1, L(ω2)⊗ L(ω2)
(q) for n = 3 and L(2ω1)⊗ L(2ω1)
(q) for n = 1.
Case II: Bn. Suppose now that G is of type Bn for n > 2. We need to consider
the p-restricted modules L(ω1) for n > 2 which has dimension 2n + 1 − δ2p, L(ωn) for
n ∈ [2, 6] which has dimension 2n, the module L(ω2) for n > 3 which has dimension
n(2n+ 1) = dimG for p 6= 2 and at least 2n(n− 1) for p = 2 (this is the adjoint module
for n > 3) and L(2ω2) for n = 2 6= p which has dimension 10.
We calculate lower bounds for the codimension of a given eigenspace for each of the
modules given. First, for the module L(ω1) we have |Λ| > rBn−1 = 1 and for the spin
module L(ωn) we have |Λ| > rAn−1 = 2
n−1. We find that |Λ| > rA1Bn−2 + rBn−1 = 2n− 1
for the module L(ω2) if p 6= 2 and |Λ| > rBn−1 = 2n − 2 if p = 2. For L(2ω2) with
n = 2 6= p the weights 2ω2 > ω1 > 0 occur, where |W.(2ω2)| = |W.ω1| = 4 and m0 = 2 so
|Λ| > rB1 + rA1 = 2 + 1 = 3.
We see that (†) holds for L(ω1)⊗ L(ω2)
(q) for n > 3 since (2n− 1− δ2p) dimL(ω1) >
dimG and for L(ω1) ⊗ L(2ω2)
(q) for n = 2 6= p since 3 dimL(ω1) = 12 > 10 = dimG.
Indeed, (†) holds for all tensor products of modules involving the adjoint modules L(ω2)
for n > 3 and L(2ω1) for n = 2 6= p. Also, (†) holds for both L(ω1) ⊗ L(ωn)
(q) and
L(ωn) ⊗ L(ωn)
(q) for n > 3 since both rAn−1 dimL(ω1) and rAn−1 dimL(ωn) = 2
2n−1
exceed dimG unless n = 2.
Hence we shall need to investigate further the modules L(ω1)⊗ L(ω2)
(q) and L(ω2)⊗
L(ω2)
(q) for n = 2, and L(ω1)⊗ L(ω1)
(q) for n > 2.
160
Case III: Cn. Suppose that G is of type Cn for n > 3, so dimG = n(2n + 1). We
need to consider the p-restricted modules L(ω1) which has dimension 2n, L(ω2) which has
dimension 2n(n− 1), the adjoint module L(2ω1) for p 6= 2, L(ωn) for n ∈ [4, 6] and p = 2
which has dimension 2n and L(ω3) for n = 3 which has dimension 8 for p = 2 and 14 for
p 6= 2 since here mω1 6= 0.
As before we calculate lower bounds for the codimension of a given eigenspace for each
of the modules given. First, for the module L(ω1) we have |Λ| > rCn−1 = 2 and for the
spin module L(ωn) we have |Λ| > rAn−1 = 2
n−2. For L(ω3) with n = 3 and p 6= 2 we have
|Λ| > 2 + 2 = 4 by including the contribution from W.ω1. The module L(ω2) satisfies
|Λ| > rA1Cn−2 = 4n−7 and for the adjoint module L(2ω1) we have |Λ| > rCn−1+rA1Cn−2 =
4n− 5.
It is clear that (†) holds for any twisted module involving L(2ω1). We see that (†)
holds for L(ω1) ⊗ L(ω2)
(q) with n > 3 since rA1Cn−2 dimL(ω1) > dimG whence also
for L(ω2) ⊗ L(ω2)
(q) with n > 3, for L(ω2) ⊗ L(ω3)
(q) with n = 3 since 2 dimL(ω2) =
24 > dimG, for L(ω2) ⊗ L(ωn)
(q) with n ∈ [4, 6] since rAn−1 dimL(ω2) > dimG and for
L(ωn) ⊗ L(ωn)
(q) with n ∈ [4, 6] since rAn−1 dimL(ωn) = 2
2n−2 > dimG. The condition
(†) holds if n = 3 and p 6= 2 for L(ω1) ⊗ L(ω3)
(q) since 4 dimL(ω1) > 21 whence for
L(ω3) ⊗ L(ω3)
(q) since dimL(ω1) < dimL(ω3) here and finally for L(ω1) ⊗ L(ωn)
(q) with
n ∈ [5, 6] and p = 2 since rAn−1 dimL(ω1) > dimG.
We shall need to investigate further the modules L(ω3)⊗L(ω3)
(q) for n = 3 and p = 2,
the module L(ω1)⊗ L(ωn)
(q) for n ∈ [3, 4] and p = 2, and L(ω1)⊗ L(ω1)
(q) for n > 3.
Case IV: Dn. Suppose that G is of type Dn for n > 4, so dimG = n(2n−1). We need
to consider the p-restricted modules L(ω1) for n > 4 which has dimension 2n, the adjoint
module L(ω2) which has dimension n(2n− 1) if p 6= 2 and n(2n− 1)− (2, n) if p = 2 and
the spin module L(ωn) for n ∈ [4, 7] with dimension 2
n−1.
We calculate that for the module L(ω1) we have |Λ| > rDn−1 = 2, for L(ω2) we have
|Λ| > rA1Dn−1 = 4n − 7 and for L(ωn) with n ∈ [4, 7] we have |Λ| > rAn−1 = 2
n−3. The
condition (†) holds for any twisted module including the adjoint module. It also holds
for L(ω1) ⊗ L(ωn)
(q) for n ∈ [6, 7] since rAn−1 dimL(ω1) > dimG. However, we need to
161
further investigate the modules L(ω1) ⊗ L(ωn−1)
(q) and L(ω1) ⊗ L(ωn)
(q) for n ∈ [4, 5],
and L(ω1)⊗ L(ω1)
(q) for n > 4.
Case V: Exceptional types. If G = (E6)sc(K) then we need to consider the p-restricted
modules L(ω1) (and L(ω6)) which has dimension 27 and the adjoint module L(ω2) which
has dimension 78 − δ3p. We see that for L(ω1) we have |Λ| > rD5 = 6 and for L(ω2) we
have |Λ| > rA5 = 21. It is clear from Proposition 4.3 since dimG = 78 that (†) holds for
all twisted modules.
If G = (E7)sc(K) then we need to consider the p-restricted modules L(ω7) which has
dimension 56 and the adjoint module L(ω1) which has dimension 133 − δ2p. For L(ω7)
we have |Λ| > rE6 = 12 and for L(ω2) we have |Λ| > rD6 = 33. Again, it is clear from
Proposition 4.3 since dimG = 133 that (†) holds in all cases here.
If G = E8(K) then we need only to consider the adjoint module L(ω8) which has
dimension 248. We have |Λ| > rE7 = 57 so we are done immediately by Proposition 4.3.
If G = F4(K) then we need to consider L(ω4) which has dimension 26 − δ3p and the
adjoint module L(ω1) which has dimension 52 if p 6= 2 and 26 if p = 2. We calculate
rB3 = 9 and rC3 = 6 so we are done.
Finally, if G = G2(K) then the modules to be considered are V1 = L(ω1) which has
dimension 7− δ2p and V2 = L(ω2) which has dimension 7 if p = 3 and 14 if p 6= 3. We find
that |Λ1| > rA1 = 3 and |Λ2| > rA˜1 = 2 so (†) holds for V1⊗V
(q)
1 since |Λ1| dimV1 > dimG
and for V1 ⊗ V
(q)
2 since |Λ1| dimV2 > dimG. The module V2 ⊗ V
(q)
2 requires further
consideration if p = 3 since |Λ2| dimV2 = 2.7 = 14 here; if p 6= 3 then (†) holds.
We list in Table 10.1 the irreducible twisted modules which require further considera-
tion.
10.2 Weight net analysis
We shall begin with the modules L(ω1)⊗ L(ωn−1)
(q) and L(ω1)⊗ L(ωn)
(q) for n ∈ [4, 5].
162
Type Module n Lemma
An L(ω1)⊗ L(ω2)
(q) [7,∞) 10.5 (†)
[3, 6] 10.5 (♦)
L(ω1)⊗ L(ωn−1)
(q) [7,∞) 10.5 (†)
[3, 6] 10.5 (♦)
L(ωn)⊗ L(ω2)
(q) [7,∞) 10.5 (†)
[3, 6] 10.5 (♦)
L(ωn)⊗ L(ωn−1)
(q) [7,∞) 10.5 (†)
[3, 6] 10.5 (♦)
L(ω1)⊗ L(2ω1)
(q) [3,∞) 10.5 (†)
[1, 2] 10.5 (♦)
L(ω1)⊗ L(2ωn)
(q) [3,∞) 10.5 (†)
[1, 2] 10.5 (♦)
L(ωn)⊗ L(2ω1)
(q) [3,∞) 10.5 (†)
[1, 2] 10.5 (♦)
L(ωn)⊗ L(2ωn)
(q) [3,∞) 10.5 (†)
[1, 2] 10.5 (♦)
L(ω2)⊗ L(ω2)
(q) 3 10.6 (†)
L(2ω1)⊗ L(2ω1)
(q) 1 10.6 (†)
Bn L(ω1)⊗ L(ω1)
(q) [2,∞) -
L(ω1)⊗ L(ω2)
(q) 2 10.6 (♦)
L(ω2)⊗ L(ω2)
(q) 2 10.6 (♦)
Cn L(ω1)⊗ L(ω1)
(q) [3,∞) -
L(ω1)⊗ L(ωn)
(q) [3, 4] (p = 2) 10.6 (†)
L(ω3)⊗ L(ω3)
(q) 3 (p = 2) 10.6 (†)
Dn L(ω1)⊗ L(ω1)
(q) [4,∞) -
L(ω1)⊗ L(ω5)
(q) 5 10.3 (†)
L(ω1)⊗ L(ω4)
(q) 5 10.3 (†)
L(ω1)⊗ L(ω4)
(q) 4 10.3 (♦)
L(ω1)⊗ L(ω3)
(q) 4 10.3 (♦)
G2 L(ω2)⊗ L(ω2)
(q) (p = 3) 10.4 (†)
Table 10.1: Possible twisted modules for all Lie types
Lemma 10.3. Let G = Spin2n(K) act on L(ω1) ⊗ L(ωn−1)
(q) and L(ω1) ⊗ L(ωn)
(q) for
n ∈ [4, 5]. In both cases, if n = 5 then (†) holds and if n = 4 then (♦) holds for all X ∈ I.
Proof. First consider the module L(ω1). There are 2(n − 1)(n − 2) or 2(n − 1) roots
α ∈ Φ(Dn) according as 〈ω1, α〉 = 0 or 1. Thus, for a given α there are
2n.2(n−1)(n−2)
2n(n−1)
=
2(n − 2) weights µ ∈ W.ω1 with weight string µ and similarly 2 weights µ ∈ W.ω1 with
163
weight string µ µ+ α.
For the module L(ωn) there are n(n− 1) or
1
2
n(n− 1) roots α ∈ Φ(Dn) according as
〈ωn, α〉 = 0 or 1. Thus, for a given α there are
2n−1.n(n−1)
2n(n−1)
= 2n−2 weights µ ∈ W.ωn with
weight string µ and similarly 2n−3 weights µ ∈W.ωn with weight string µ µ+ α.
By combining these two calculations, given α /∈ Φs the weight nets for L(ω1)⊗L(ωn)
(q)
where µ ∈ W.ω1 and µ
′ ∈ W.ωn are given below along with the number of each type of
weight net.
µ+ qµ′ µ+ qµ′ µ+ q(µ′ + α) µ+ qµ′
(µ+ α) + qµ′
µ+ qµ′ µ+ q(µ′ + α)
(µ+ α) + qµ′ (µ+ α) + q(µ′ + α)
2n−1(n− 2) 2n−2(n− 2) 2n−1 2n−2
Thus we have |Λ| > 2n−2(n− 2) + 2n−1 + 2n−1 = 2n−2(n+ 2) which exceeds dimG for
n = 5 so (†) holds here and equals |Φ(Dn)| = 24 for n = 4 in which case we are left to
show that (♦) holds for X = ∅.
The 8 weights in W.ω1 are as follows.
1 1 1
2
1
2
0 1 1
2
1
2
0 0 1
2
1
2
0 0 1
2
- 1
2
0 0 - 1
2
1
2
0 0 - 1
2
- 1
2
0 -1 - 1
2
- 1
2
-1 -1 - 1
2
- 1
2
Similarly, the 8 weights in W.ω4 are as follows.
1
2
1 1
2
1 1
2
1 1
2
0 1
2
0 1
2
0 - 1
2
0 1
2
0 1
2
0 - 1
2
0 - 1
2
0 - 1
2
0 - 1
2
-1 - 1
2
0 - 1
2
-1 - 1
2
-1
If we write the 64 weights µ + qµ′ where µ ∈ W.ω1, µ
′ ∈ W.ω4 of L(ω1) ⊗ L(ω4)
(q)
in an 8 × 8 grid then it is clear that in each row and each column at least 6 weights lie
in Λ; thus d∅ω1+qω4 = 48 > 24 = e
∅
ω1+qω4 . The lemma follows for L(ω1) ⊗ L(ωn−1)
(q) with
n ∈ [4, 5] similarly.
In the next lemma we shall deal with the only twisted module requiring consideration
for a group of exceptional type.
Lemma 10.4. Consider the simple algebraic group G = G2(K) acting on L(ω2)⊗L(ω2)
(q)
with p = 3. Then (†) holds.
Proof. When p = 3 the weights in W.ω1 appear with multiplicity zero and the weight 0
appears with multiplicity one in L(ω2). The short roots in Φ(G2) are ±α1, ±(α1 + α2)
and ±(2α1 + α2) and we may assume that one of these does not lie in Φs. We see that
〈ω2, α〉 = 0 for α ∈ {±α1} and 〈ω2, α〉 = 3 for α ∈ {α1 + α2, 2α1 + α2}. Thus, given a
164
short root α, there are 6.2
6
= 2 weights µ ∈ W.ω2 with 〈µ, α〉 = 0 and similarly 2 weights
µ ∈ W.ω2 with 〈µ, α〉 = 3. If p 6= 3 there are two weight strings of the form µ2 µ1 µ1 µ2
where µi ∈ W.ωi for i = 1, 2. Although mω1 = 0 we may still use the adjacency principle
for p = 3 by taking α /∈ Φs and using the fact that µ(s) = (µ − 3α)(s) if and only if
α(s) = 1. If µ, µ′ ∈ W.ω2 then the weight nets that occur for L(ω2) ⊗ L(ω2)
(q) are as
follows.
µ+ qµ′ µ+ qµ′ µ+ q(µ′ + 3α) µ+ qµ′
(µ+ 3α) + qµ′
µ+ qµ′ µ+ q(µ′ + 3α)
(µ+ 3α) + qµ′ (µ+ 3α) + q(µ′ + 3α)
4 4 4 4
Thus we have |Λ| > 4.1 + 4.1 + 4.2 = 16 > dimG = 14 as required.
Recall from Section 5.3 that for type An the weights in W.ωk with k ∈ [1, n] can be
represented by strings of length n+1 consisting of k plus signs and n−k+1 minus signs.
Also elements of the Weyl group act by permuting plus and minus signs.
Lemma 10.5. Let G = SLn+1(K) act on U ⊗ V
(q) where U ∈ {L(ω1), L(ωn)} and V ∈
{L(ω2), L(ωn−1)} for n ∈ [3,∞), and U ⊗ V
(q) where U ∈ {L(ω1), L(ωn)} and V ∈
{L(2ω1), L(2ωn)} for n ∈ [1,∞). Then in the former case (†) holds for n ∈ [7,∞) and
(♦) is always satisfied for n ∈ [3, 6], and in the latter case (†) holds for n ∈ [3,∞) and
(♦) is always satisfied for n ∈ [1, 2].
Proof. We begin by recalling that dimL(ω1) = n + 1, dimL(ω2) =
1
2
n(n + 1) and
dimL(2ω1) =
1
2
(n + 1)(n + 2). Following the usual calculations we find that, for a given
α ∈ Φ(An), there are n− 1 weights µ ∈ W.ω1 with weight string µ and there is 1 weight
µ ∈ W.ω1 with weight string µ µ + α. Similarly, there are 1 +
1
2
(n − 2)(n − 1) weights
ν ∈W.ω2 with weight string ν and there are n− 1 weights ν ∈ W.ω2 with weight string ν
ν+α. Finally, considering L(2ω1), there are n−1 weights η ∈W.(2ω1) with weight string
η and there is 1 weight η ∈ W.(2ω1) with weight string η η+α η+2α where η+α ∈ W.ω2.
Consider the module L(ω1)⊗L(ω2)
(q); taking α /∈ Φs we count the number of weights
that occur as singletons, i.e., those weights of the form µ + qν with 〈µ, α〉 = 〈ν, α〉 = 0.
There are (n−1)+ 1
2
(n−2)(n−1)2 such weights. The remaining weights of L(ω1)⊗L(ω2)
(q)
165
all occur in pairs so we find that |Λ| > 1
2
(
1
2
n(n+ 1)2 − (n− 1)− 1
2
(n− 2)(n− 1)2
)
=
1
2
(3n2 − n+ 2). This exceeds dimG for n > 7 so (†) holds.
We may now take two orthogonal roots outside the root system Φs of CG(s) since (♦)
is satisfied if X = An−1 for n > 3. Assume without loss that these two roots are α1 and
α3. We see that there are
(
n−3
1
)
= n − 3 weights in W.ω1 orthogonal to both of these
roots and similarly there are 2 +
(
n−3
2
)
weights in W.ω2 orthogonal to them both (those
weights of the form ++−−− · · ·−, −−++− · · ·− and −−−−++− · · ·−). Thus we now find that
|Λ| > 1
2
(
1
2
n(n+ 1)2 − 2(n− 3)− 1
2
(n− 4)(n− 3)2
)
= 3n2− 9n+12 which exceeds dimG
for n > 5. If n = 4 then |Λ| > 24 = dimG so (♦) is satisfied for all X ∈ I and if n = 3
then |Λ| > 12 = Φ(A3) so we are left to consider the possibility X = ∅ here.
Now if n = 3 then the weights in W.ω1 are 3 2 1, -1 2 1, -1 -2 1 and -1 -2 -3 where
we have omitted a factor of 1
4
on each coefficient. Clearly the difference between any two
of these is a root in A3. Suppose that X = ∅; if we arrange the 24 weights µ + qν with
µ ∈ W.ω1, ν ∈ W.ω2 of L(ω1) ⊗ L(ω2)
(q) into a 6 × 4 grid then, in each column, at least
three of the four weights lies in Λ. Thus d∅ω1+qω2 > 18 > 12 = e
∅
ω1+qω2 . (We are essentially
using Proposition 4.3 here.)
It is clear that the same conclusions for L(ω1) ⊗ L(ω2)
(q) with n ∈ [3,∞) hold for
L(ω1)⊗ L(ωn−1)
(q), L(ωn)⊗ L(ω2)
(q) and L(ωn)⊗ L(ωn−1)
(q) with n ∈ [3,∞).
Consider the module L(ω1)⊗L(2ω1)
(q). Since ω2 < 2ω1 we can use Premet’s theorem
to conclude that (†) holds for n > 7. Using the data at the start of this lemma we find
that the weight nets for this module are as follows.
µ+ qν µ+ qν µ+ q(ν + α) µ+ q(ν + 2α) µ+ qν
(µ+ α) + qν
µ+ qν µ+ q(ν + α) µ+ q(ν + 2α)
(µ+ α) + qν (µ+ α) + q(ν + α) (µ+ α) + q(ν + 2α)
(n− 1)2 n− 1 n− 1 1
µ+ qη µ+ qη µ+ q(η + α) µ+ qη
(µ+ α) + qη
µ+ qη µ+ q(η + α)
(µ+ α) + qη (µ+ α) + q(η + α)
1
2
(n− 2)(n− 1)2 (n− 1)2 1
2
(n− 2)(n− 1) n− 1
Thus we have |Λ| > 1
2
(3n2 + n + 2) which exceeds dimG for n > 3 and equals dimG
for n ∈ [1, 2].
We can draw the same conclusions when n ∈ [2,∞) for L(ωn) ⊗ L(2ω1)
(q), L(ω1) ⊗
166
L(2ωn)
(q) and L(ωn)⊗ L(2ωn)
(q).
The last lemma of this chapter deals with the remaining six families of modules in
Table 10.1.
Lemma 10.6. Let G be a simple simply connected algebraic group defined over K.
(i) If G is of type An acting on either L(ω2)⊗L(ω2)
(q) for n = 3 or L(2ω1)⊗L(2ω1)
(q)
for n = 1 then (†) holds.
(ii) If G is of type B2 acting on either L(ω1)⊗ L(ω2)
(q) or L(ω2)⊗ L(ω2)
(q) then (♦) is
satisfied for all X ∈ I.
(iii) If G is of type Cn with p = 2 acting on either L(ω1) ⊗ L(ωn)
(q) for n ∈ [3, 4] or
L(ω3)⊗ L(ω3)
(q) for n = 3 then (†) holds.
Proof. Consider the action of SL2(K) on L(2ω1)⊗L(2ω1)
(q). We can arrange the 9 weights
of this module in a 3× 3 grid as follows.
2ω1 + q(2ω1) 2ω1 + q0 2ω1 − q(2ω1)
0 + q(2ω1) 0 + q0 0− q(2ω1)
−2ω1 + q(2ω1) −2ω1 + q0 −2ω1 − q(2ω1)
Clearly, by observing that 2ω1 = α1, we have |Λ| > 4 > 3 = dimG so (†) holds.
Consider the action of SL4(K) on L(ω2) ⊗ L(ω2)
(q). If µ, µ′ ∈ W.ω2 and α /∈ Φs
then, using the calculations in the previous lemma, the weight nets for this module are
as follows.
µ+ qµ′ µ+ qµ′ µ+ q(µ′ + α) µ+ qµ′
(µ+ α) + qµ′
µ+ qµ′ µ+ q(µ′ + α)
(µ+ α) + qµ′ (µ+ α) + q(µ′ + α)
4 4 4 4
Thus |Λ| > 4 + 4 + 8 = 16 > 15 = dimG.
Consider the action of Spin5(K) on L(ω1)⊗ L(ω2)
(q). Recall that dimL(ω2) = 4 and
dimL(ω1) = 5 or 4 according as p 6= 2 or p = 2 (the difference in the latter case is that
m0 = 0). For a fixed short root α ∈ Φ(B2) there are 2 weights µ ∈ W.ω1 with weight
string µ and 1 weight µ ∈ W.ω1 with weight string µ µ + α µ + 2α where the middle
weight in the string is 0. Similarly we find that there are 2 weights µ′ ∈ W.ω2 with weight
string µ′ µ′ + α.
167
Assuming that α /∈ Φs we can arrange the weights of L(ω1)⊗ L(ω2)
(q) into four 2× 1
and two 2×3 or 2×2 weight nets according as p 6= 2 or p = 2. If the characteristic is odd
then |Λ| > 4.2 + 2.3 = 10 = dimG and if it is even then, using the fact that 2α(s) = 1 if
and only if α(s) = 1, we have |Λ| > 4 + 2.2 = 8 = |Φ(B2)|. It remains to show that (♦)
holds for X = ∅ if p = 2. The weights of L(ω1) are 1 1, 0 1, 0 -1, -1 -1 and the weights of
L(ω2) are
1
2
1, 1
2
0, -1
2
0 and -1
2
-1 so if we arrange the 16 weights of L(ω1)⊗ L(ω2)
(q) for
p = 2 into a 4× 4 grid then at least three weights in each row and column lie in Λ. Thus
d∅ω1+qω2 > 4.3 = 12 > 8 = e
∅
ω1+qω2 .
Now consider the action of Spin5(K) on L(ω2) ⊗ L(ω2)
(q). Taking µ, µ′ ∈ W.ω2 and
α /∈ Φs we find |Λ| > 8 since there are four 2 × 2 weight nets for this module. We just
need to consider the case X = ∅ to show that (♦) holds for all X ∈ I, but this follows
immediately as before.
Consider the action of Sp2n(K) with p = 2 on L(ω1) ⊗ L(ωn)
(q) for n ∈ [3, 4]. Then
dimL(ω1) = 2n and dimL(ωn) = 2
n. We have 〈ωn, α〉 = 0 or 2 for either n(n − 1) or
1
2
n(n − 1) short roots α ∈ Φ(Cn). For a given α there are 2
n−1 weights µ ∈ W.ωn with
weight string µ and 2n−2 weights µ ∈ W.ωn with weight string containing µ µ + 2α. For
L(ω1) there are 2(n− 2) weights µ
′ ∈W.ω1 with weight string µ
′ and 2 weights µ′ ∈W.ω1
with weight string µ′ µ′ + α.
The weight nets for L(ω1)⊗ L(ωn)
(q) with p = 2 are as follows where α /∈ Φs.
µ+ qµ′ µ+ qµ′ µ+ q(µ′ + 2α) µ+ qµ′
(µ+ α) + qµ′
µ+ qµ′ µ+ q(µ′ + 2α)
(µ+ α) + qµ′ (µ+ α) + q(µ′ + 2α)
2n(n− 2) 2n−1(n− 2) 2n 2n−1
Therefore we have |Λ| > 2n−1(n− 2) + 2n+2n = 2n−1(n+2) which exceeds dimG for
n = 4. If n = 3 then we have |Λ| > 20 whereas dimG = 21 so more work is required.
We need to show that dXω1+qω3 > dimG for X = A2 and X = C2 in order to take two
orthogonal roots (one short and one long) outside Φs and conclude that (†) holds. Suppose
X = A2 with roots α1 and α2. The weights in W.ω1 split into a clique consisting of two
clusters · · 1
2
and · · -1
2
of size 3. By the adjacency principle for L(ω1) at least one of these
clusters does not lie in the eigenspace. The weights in W.ω3 split into four clusters 1 2
3
2
,
· · 1
2
, · · -1
2
and -1 -2 -1
2
of sizes 1, 3, 3 and 1 respectively. Thus, by the adjacency principle
168
for L(ω3) with p = 2, the maximum number of weights that can lie in the eigenspace is 3
(from either cluster of size 3). We form a 6× 8 grid consisting of weights µ + qµ′ where
µ ∈ W.ω1 and µ
′ ∈W.ω3. Arranging these weights into eight clusters (four of the form 3×1
and four of the form 3× 3) we find that dω1+qω3 > 2(1.3 + 3.3 + 1.3) = 30 > 21 = dimG.
Suppose that X = C2 with roots α2 and α3. The weights in W.ω1 split into a clique
consisting of three clusters 1 1 1
2
, 0 · · and -1 -1 -1
2
of sizes 1, 4 and 1 respectively. The
weights in W.ω3 split into a clique consisting of two clusters 1 · · and -1 · ·, both of size
4. As before, we form a 6 × 8 grid consisting of weights µ + qµ′ where µ ∈ W.ω1 and
µ′ ∈ W.ω3. Arranging these weights into six clusters (four of the form 1 × 4 and two of
the form 4× 4) we find that dω1+qω3 > 4 + 4 + 16 + 4 = 28 > 21 = dimG.
Now we take two orthogonal roots α short and β long outside Φs. We see that 〈ω1, α〉 =
0 or ±1 and 〈ω1, β〉 = 0 or ±1; however it is not possible to find two orthogonal roots α,
β such that 〈ω1, α〉 = 〈ω1, β〉 = 0 since n = 3. Similarly we have 〈ω3, α〉 = 0 or ±2 (in the
latter case the weight strings obtained are not saturated) and 〈ω3, β〉 = ±1. In particular
there are no weights in both L(ω1) and L(ω3) orthogonal to both a short and long root
orthogonal to each other. Thus the weights in L(ω1) ⊗ L(ω3)
(q) for n = 3 all appear in
pairs when arranged into weight nets for some fixed pair of mutually orthogonal short
and long roots, whence
|Λ| >
1
2
dimL(ω1)⊗ L(ω3)
(q) = 24 > 21 = dimG
and (†) holds.
Finally, consider the action of Sp6(K) with p = 2 on L(ω3) ⊗ L(ω3)
(q). The weight
nets that occur are similar to those in the previous case except in the vertical direction
the difference between two weights is 2α rather than α. We find |Λ| > 8 + 8 + 2.4 = 24,
so (†) holds.
It remains to consider L(ω1)⊗ L(ω1)
(q) for types Bn, Cn and Dn. We should be able
to adapt our techniques to show for each n that (♦) is satisfied for all X ∈ I. The natural
approach is to inductively take mutually orthogonal roots outside Φs.
169
Chapter 11
Concluding remarks
In this final chapter we shall briefly discuss a number of points that have arisen in the
course of this thesis.
The Main Theorem presents a clear dichotomy between modules of dimension at most
and larger than dimG + 2. It is somewhat surprising and not entirely clear that this
should be the dividing line. Indeed, one would perhaps intuitively expect that dimG is a
more likely boundary. There are very few modules with dimension dimG+1 or dimG+2,
the most notable is the infinite family L(ω1) ⊗ L(ω1)
(q) in type An for n > 1; the other
modules occur only for a single rank n for small characteristic, or else for n = 1.
Our methodology has been to begin each chapter on large modules with an initial sur-
vey to decide which modules we cannot yet conclude that (†) holds. We have a reasonably
efficient technique to do this and we show that (†) holds for the vast majority of modules.
It is only for modules with dimension closer to dimG+2 that more work is required and
we usually show that (♦) is satisfied for each X ∈ I rather than that (†) holds. Indeed,
if a module has dimension approximately 2 dimG then we expect to be able to show at
least that (♦) is satisfied for all X ∈ I.
In some cases we have been able to assume that there are two or more orthogonal
roots lying outside the root system Φs for some s ∈ Gss \ Z since (♦) is satisfied for
sufficiently large centraliser types. In these cases, it is subsequently possible to show that
the codimension of the eigenspace exceeds dimG. However, we cannot conclude that (†)
170
holds since there are simply too many large centraliser types X to check to ensure that
dXλ > dimG always holds.
It would be reasonable to expect that a computer program could be produced to find,
for any possible centraliser type, better lower bounds for the codimension of an eigenspace.
This would be easier in cases where the plus minus notation for weights can be used, i.e.,
for modules L(ωk) for k ∈ [1, n] in type An, and L(ωn) for types Bn, Cn (p = 2 only) and
Dn. It is far from efficient when there are many centraliser types to check by hand to
show that (♦) is satisfied for all X ∈ I. For example, there were 60 centraliser types to
go through for the module L(ω7) for type D7 with p = 2.
We have not made any claims about the size of the codimension of the eigenspace for
modules where (†) holds; for modules V where dimV > 3 dimG the codimension of the
eigenspace would certainly far exceed dimG with few exceptions. We do not claim that
(†) is an optimal condition, rather that it is convenient. It allows us to reach our desired
conclusion that dimE < dimV .
Naturally, we would like to have fewer possible exceptions to the Main Theorem. In
particular there are three infinite families for which our methods do not appear to be
sufficient: L(ω2) for n ∈ [3,∞) in type Cn, L(2ω1) for n ∈ [2,∞) with p 6= 2 in type
Bn and L(2ω1) for n ∈ [4,∞) with p 6= 2 in type Dn. It should be possible to deal
with the modules L(ω1)⊗ L(ω1)
(q) for types Bn, Cn and Dn; an approach by inductively
taking orthogonal roots not in Φs should show that (♦) is satisfied for all X ∈ I. Other
modules with dimension at most dimG could possibly be dealt with using the strategy
employed by Lawther in Section 3.6, though this would be most difficult in the case L(ω3)
for SL8(K). Many of the remaining modules with dimension larger than dimG + 2 can
be dealt with for almost all centraliser types X ∈ I. In cases where dXλ = e
X
λ occurs, it
should be possible to say more since we have quite a bit of information. We know the
centraliser type of a given non-central semisimple element and precisely which weights lie
in the eigenspace.
We have shown that the set of vectors whose stabilisers contain no non-central semisim-
ple element is dense in V . It remains to show that at least one of these vectors has a
171
stabiliser containing no non-trivial unipotent element. Clearly this will require a different
strategy to that employed in this thesis for non-central semisimple elements. There has
been much work done by Suprunenko on the behaviour of unipotent elements in modular
representations of algebraic groups which may be helpful in this regard; she intends to
obtain estimates for the dimensions of the 1-eigenspaces V u = {v ∈ V | uv = v} for each
unipotent u ∈ G following on from the work in [28].
It would be helpful to use the existence of a regular orbit for the action of a simple
algebraic group to deduce that a primitive permutation group of affine type has minimal
base size 2. Indeed, much work has already been done on the almost simple permutation
groups, particularly by Burness.
It should be possible to find further applications of the Main Theorem. In particular,
we have shown for the majority of modules with dimension exceeding dimG+ 2 that (†)
holds. This may be useful for problems on eigenvectors in actions of simple algebraic
groups, some of which are described in [29].
Dyfal donc a dyr y garreg
N´ı heolas go haontios
172
Appendix A
Cliques for (G, V ) = (A12, L(ω3))
We shall provide cliques for the nineteen centraliser types remaining to be considered for
SL13(K) acting on L(ω3) listed in Table 5.2 of Lemma 5.13. We show in each case that
(♦) is satisfied.
X = ∅
Clique Cluster size l Clique Cluster size l
+++−−−−−−−−−− 1 60 −−+++−−−−−−−− 1 40
−+++−−−−−−−−− 1 50 −−−+++−−−−−−− 1 32
From the table above we have d∅ω3 > 182 > 156 = e
∅
ω3
.
X = A1
Clique Cluster size l
+− : + +−−−−−−−−− 2 100
−− : ++ +−−−−−−−− 1 40
−− : −+++−−−−−−− 1 32
From the table above we have dA1ω3 > 172 > 154 = e
A1
ω3
.
X = A2
1
Clique Cluster size l
+− |+− : +−−−−−−−− 4 32
+− | − − : + +−−−−−−− 2 64
−− |+− : + +−−−−−−− 2 64
From the table above we have d
A2
1
ω3 > 160 > 152 = e
A2
1
ω3 .
173
X = A3
1
Clique Cluster size l Clique Cluster size l
+− |+−| − − : +−−−−−− 4 24 +− | − −| − − : + +−−−−− 2 36
+− | − −|+− : +−−−−−− 4 24 −− |+−| − − : + +−−−−− 2 36
−− |+−|+− : +−−−−−− 4 24 −− | − −|+− : + +−−−−− 2 36
From the table above we have d
A3
1
ω3 > 180 > 150 = e
A3
1
ω3 .
X = A4
1
Clique Cluster size l Clique Cluster size l
+− | − −| − −| − − : + +−−− 2 16 +− | − −|+−| − − : +−−−− 4 16
−− |+−| − −| − − : + +−−− 2 16 +− | − −| − −|+− : +−−−− 4 16
−− | − −|+−| − − : + +−−− 2 16 −− |+−|+−| − − : +−−−− 4 16
−− | − −| − −|+− : + +−−− 2 16 −− |+−| − −|+− : +−−−− 4 16
+− |+−| − −| − − : +−−−− 4 16 −− | − −|+−|+− : +−−−− 4 16
From the table above we have d
A4
1
ω3 > 160 > 148 = e
A4
1
ω3 .
X = A5
1
Clique Cluster size l Clique Cluster size l
+− |+−|+−| − −| − − : −−− 8 64 +− |+−| − −| − −| − − : −+− 4 32
+− |+−| − −| − −| − − : +−− 4 32 +− |+−| − −| − −| − − : −−+ 4 32
From the table above we have d
A5
1
ω3 > 160 > 146 = e
A5
1
ω3 .
X = A6
1
Clique Cluster size l Clique Cluster size l
+− |+−|+−| − −| − −| − − : − 8 128 +−|++|− − | − −| − −| − − : − 2 9
+ + |+− | − −| − −| − −| − − : − 2 9 −− |++| − −| − −| − −| − − : + 1
+ + | − −| − −| − −| − −| − − : + 1
From the table above we have d
A6
1
ω3 > 146 > 144 = e
A6
1
ω3 .
X = A2
Clique Cluster size l
++− : +−−−−−−−−− 3 27
+−− : + +−−−−−−−− 3 120
−−− : + + +−−−−−−− 1 7
From the table above we have dA2ω3 > 154 > 150 = e
A2
ω3
.
X = A2A1
Clique Cluster size l
+−−|+− : +−−−−−−− 6 42
+−−| − − : + +−−−−−− 3 72
−−−|+− : + +−−−−−− 2 48
174
From the table above we have dA2A1ω3 > 162 > 148 = e
A2A1
ω3
.
X = A2A21
Clique Cluster size l Clique Cluster size l
++−| − −|+− : −−−−−− 6 42 +−−| − −| − − : + +−−−− 3 36
+−−|+−|+− : −−−−−− 12 −−−|+−| − − : + +−−−− 2 24
+−−|+−| − − : +−−−−− 6 −−−| − −|+− : + +−−−− 2 24
+−−| − −|+− : +−−−−− 6 30
From the table above we have d
A2A21
ω3 > 156 > 146 = e
A2A21
ω3 .
X = A2A31
Clique Cluster size l Clique Cluster size l
+−−|+− |+−| − − : −−−− 12 32 + +−|+− | − −| − − : −−−− 6 24
−−−|+−|+−|+− : −−−− 8 + +−| − −| − −| − − : +−−− 3
+−−|+−| − −| − − : +−−− 6 18 −−−|+−|+−| − − : +−−− 4 12
+−−| − −|+−| − − : +−−− 6 18 −−−|+−| − −|+− : +−−− 4 12
+−−| − −| − −|+− : +−−− 6 18 −−−| − −|+−|+− : +−−− 4 12
From the table above we have d
A2A31
ω3 > 146 > 144 = e
A2A31
ω3 .
X = A2A41
Clique Cluster size l Clique Cluster size l
++−|+− | − −| − −| − − : −− 6 24 +−−|+− |+−| − −| − − : −− 12 48
+ +−| − −| − −| − −| − − : +− 3 −−−|+− |+−|+−| − − : −− 8 24
+−−|+− | − −| − −| − − : +− 6 18 −−−|+− |+−| − −| − − : +− 4 16
+−−|+− | − −| − −| − − : −+ 6 18
From the table above we have d
A2A41
ω3 > 148 > 142 = e
A2A41
ω3 .
X = A2A51
Clique Cluster size l
+−−|+− |+−| − −| − −| − − : 12 96
−−−|+− |+−|+−| − −| − − : 8 64
From the table above we have d
A2A51
ω3 > 160 > 140 = e
A2A51
ω3 .
X = A2
2
Clique Cluster size l
+−−|+−− : +−−−−−− 9 54
+−−| − −− : + +−−−−− 3 54
−−−|+−− : + +−−−−− 3 54
From the table above we have d
A2
2
ω3 > 162 > 144 = e
A2
2
ω3 .
175
X = A2
2
A1
Clique Cluster size l Clique Cluster size l
++−|+−− | − − : −−−−− 9 63 +−−| − −− |+− : +−−−− 6 24
+−−|++− | − − : −−−−− 9 −−−|+−− |+− : +−−−− 6 24
+−−|+−− |+− : −−−−− 18 +−−| − −− | − − : + +−−− 3 24
+−−|+−− | − − : +−−−− 9 −−−|+−− | − − : + +−−− 3 24
From the table above we have d
A2
2
A1
ω3 > 159 > 142 = e
A2
2
A1
ω3 .
X = A2
2
A2
1
Clique Cluster size l Clique Cluster size l
+−−|+−− |+− | − − : −−− 18 45 +−−| − −−| − −| − − : +−+ 3 7
+−−|+−− | − −| − − : +−− 9 −−−| − −− |+− | − − : +−+ 2
+−−| − −− |+−| − − : +−− 6 12 +−−| − −−| − −| − − : −++ 3 7
+−−| − −− | − −|+− : +−− 6 12 −−−| − −− |+− | − − : −++ 2
−−−|+−− |+−| − − : +−− 6 12 + +−| − −− |+− | − − : −−− 6 15
−−−|+−− | − −|+− : +−− 6 12 + +−| − −− | − −| − − : +−− 3
+−−| − −−| − −| − − : + +− 3 7 −−−|++− |+− | − − : −−− 6 15
−−−| − −− |+− | − − : + +− 2 −−−|++− | − −| − − : +−− 3
From the table above we have d
A2
2
A2
1
ω3 > 144 > 140 = e
A2
2
A2
1
ω3 .
X = A3
Clique Cluster size l
++−− : +−−−−−−−− 6 48
+−−− : + +−−−−−−− 4 128
From the table above we have dA3ω3 > 176 > 144 = e
A3
ω3
.
X = A3A1
Clique Cluster size l
++−− | − − : +−−−−−− 6 36
+−−− |+− : +−−−−−− 8 48
+−−− | − − : + +−−−−− 4 72
From the table above we have dA3A1ω3 > 156 > 142 = e
A3A1
ω3
.
X = A3A21
Clique Cluster size l Clique Cluster size l
++−− |+−| − − : −−−−− 12 56 + +−− | − −|+− : −−−−− 12 44
+−−− |++| − − : −−−−− 4 +−−− | − −|++ : −−−−− 4
+−−− |+−|+− : −−−−− 16 +−−− | − −|+− : +−−−− 8
+−−− |+−| − − : +−−−− 8 + ++− | − −| − − : −−−−− 4 28
−−−− |+−|+− : +−−−− 4 16 + +−− | − −| − − : +−−−− 6
From the table above we have d
A3A21
ω3 > 144 > 140 = e
A3A21
ω3 .
176
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