On Problems in the Representation Theory of Symmetric Groups

Abstract

In this thesis, we study the representation theory of the symmetric groups $\mathfrak{S}_n$, their Sylow $p$-subgroups $P_n$ and related algebras. For all primes $p$ and natural numbers $n$, we determine the maximum number of distinct irreducible constituents of degree coprime to $p$ of restrictions of irreducible characters of $\mathfrak{S}n$ to $\mathfrak{S}{n-1}$, and show that every value between 1 and this maximum is attained. These results can be stated graph-theoretically in terms of the Young lattice, which describes branching for symmetric groups. We present new graph isomorphisms between certain subgraphs of the Young lattice and find self-similar structures. This generalises from $p=2$ to all $p$ work of Ayyer, Prasad and Spallone which was central in the construction of character correspondences for symmetric groups in the context of the McKay Conjecture, a fundamental open problem in the representation theory of finite groups. Linear characters of Sylow subgroups have also played a central role in character correspondences verifying the McKay Conjecture, becoming the focus of much current interest. For instance, a consequence of recent work of Giannelli and Navarro shows the existence of linear constituents in the restriction of every irreducible character of a symmetric group to its Sylow $p$-subgroups. We now identify these linear constituents, using a mixture of algebraic and combinatorial techniques including Mackey theory and an analysis of Littlewood--Richardson coefficients. We determine precisely when the trivial character of $P_n$ appears as a constituent of the restriction of an irreducible character of $\mathfrak{S}_n$, for all $n$ and odd $p$. As a consequence, we determine the irreducible characters of the Hecke algebra corresponding to the induced permutation character. Analogous results are obtained for the alternating groups $\mathfrak{A}_n$. We then extend our scope to arbitrary linear characters of $P_n$, proving in particular that for all $p$, given linear characters $\phi$ and $\phi'$ of $P_n$, their inductions to $\mathfrak{S}n$ are equal if and only if $\phi$ and $\phi'$ are $N{\mathfrak{S}_n}(P_n)$--conjugate. Finally, we consider the representation theory of Schur algebras in all characteristics. We classify the classical Schur algebras $S(n,r)$ which are Ringel self-dual, using decomposition numbers for symmetric groups, tilting module multiplicities and combinatorial methods.