dc.contributor.author Hazi, Amit dc.date.accessioned 2018-02-07T12:30:46Z dc.date.available 2018-02-07T12:30:46Z dc.date.issued 2018-02-07 dc.date.submitted 2017-09-28 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/271774 dc.description.abstract Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. The indecomposable tilting modules $\{T(\lambda)\}$ for $G$, which are labeled by highest weight, form an important class of self-dual representations over $k$. In this thesis we investigate semisimple filtrations of minimal length (Loewy series) of tilting modules. We first demonstrate a criterion for determining when tilting modules for arbitrary quasi-hereditary algebras are rigid, i.e. have a unique Loewy series. Our criterion involves checking that $T(\lambda)$ does not have certain subquotients whose composition factors extend more than one layer in the radical or socle series. We apply this criterion to show that the restricted tilting modules for $SL_4$ are rigid when $p \geq 5$, something beyond the scope of previous work on this topic by Andersen and Kaneda. Even when $T(\lambda)$ is not rigid, in many cases it has a particularly structured Loewy series which we call a balanced semisimple filtration, whose semisimple subquotients or "layers" are symmetric about some middle layer. Balanced semisimple filtrations also suggest a remarkably straightforward algorithm for calculating tilting characters from the irreducible characters. Applying Lusztig's character formula for the simple modules, we show that the algorithm agrees with Soergel's character formula for the regular indecomposable tilting modules for quantum groups at roots of unity. We then show that these filtrations really do exist for these tilting modules. In the modular case, high weight tilting modules exhibit self-similarity in their characters at $p$-power scales. This is due to what we call higher-order linkage, an old character-theoretic result relating modular tilting characters and quantum tilting characters at $p$-power roots of unity. To better understand this behavior we describe an explicit categorification of higher-order linkage using the language of Soergel bimodules. Along the way we also develop the algebra and combinatorics of higher-order linkage at the de-categorified level. We hope that this will provide a foundation for a tilting character formula valid for all weights in the modular case when $p$ is sufficiently large. dc.description.sponsorship This thesis was completed with the combined financial support of Trinity College (through an Internal Graduate Studentship) and the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. dc.language.iso en dc.rights Attribution-NonCommercial-ShareAlike 4.0 International dc.rights.uri https://creativecommons.org/licenses/by-nc-sa/4.0/ dc.subject modular representation theory dc.subject tilting modules for algebraic groups dc.subject Soergel bimodules dc.title Semisimple filtrations of tilting modules for algebraic groups dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.publisher.department Department of Pure Mathematics and Mathematical Statistics dc.date.updated 2018-02-07T12:18:12Z dc.identifier.doi 10.17863/CAM.18769 dc.contributor.orcid Hazi, Amit [0000-0001-7264-2211] dc.publisher.college Trinity College dc.type.qualificationtitle PhD in Pure Mathematics cam.supervisor Martin, Stuart rioxxterms.freetoread.startdate 2018-02-07
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