Representations of groups and Hecke algebras

A conference in memory of Anton Evseev

Tuesday 18th–Wednesday 19th September 2018

School of Mathematics

University of Birmingham

Anton
Tuesday September 18
12:30–13:30 Registration
13:30–14:20 Radha Kessar Weight conjectures for fusion systems
14:30–15:20 Rowena Paget Tableaux methods for plethysm
15:30–16:15 Coffee
16:15–17:15 Gunter Malle Representations of finite groups
17:15– Drinks reception followed by conference dinner
Wednesday September 19
9:30–10:20 Shunsuke Tsuchioka A definition of Schur regular partitions
10:30–11:00 Coffee
11:00–11:40 James Whitley Brauer Correspondence for Hecke algebras and the Dipper--Du conjecture
11:45–12:35 Sinead Lyle On bases of Specht modules corresponding to 2-column partitions
12:45–14:00 Lunch
14:00–14:50 Michael Livesey Towards Donovan's conjecture for abelian defect groups
15:00–15:40 Ryan Davies An induction theorem inspired by Brauer's induction theorem
15:45–16:15 Coffee
16:15--17:05 Joe Chuang Rank functions on triangulated categories

Titles and abstracts
Joe Chuang: Rank functions on triangulated categories
Cohn and Schofield showed that (equivalence classes of) homomorphisms from a fixed ring R into skew fields (and more generally simple Artinian rings) are parametrized by functions on finitely presented R-modules satisfying certain natural conditions, called rank functions. In this talk I will introduce a derived version of this theory and give some examples from representations of finite groups. This is joint work with Andrey Lazarev.
Ryan Davies: An induction theorem inspired by Brauer's induction theorem
Let $G$ be a finite group and $\chi$ be an ordinary irreducible character of $G$. Brauer's induction theorem states that $\chi$ can be expressed as an integer linear combination of irreducible characters of induced from elementary subgroups of $G$. My thesis aimed to prove an induction theorem inspired by Brauer's theorem which we discuss in this talk and the progress made towards the result.
Radha Kessar: Weight conjectures for fusion systems
Many of the conjectures of current interest in the representation theory of finite groups in characteristic p are local-to-global statements, in that they predict consequences about the representations of a finite group G given data about the representations of the p-local subgroups of G. The local structure of a block of a group algebra is encoded by the fusion system of the block together with a compatible family of cohomology classes. We state and initiate investigation of a number of seemingly local conjectures motivated by their counterparts in block theory. This is joint work with Markus Linckelmann, Justin Lynd, and Jason Semararo.
Michael Livesey: Towards Donovan's conjecture for abelian defect groups
Donovan's conjecture states that for a fixed defect group, up to Morita equivalence, there are only finitely many blocks with this defect group. In this talk I will present a reduction for Donovan's conjecture for abelian defect groups to blocks of quasisimple groups. In particular, this completes the proof of Donovan's conjecture for abelian defect groups in characteristic 2. The main tool in all of this is the new concept of a strong Frobenius number which is a refinement of the Morita Frobenius number introduced by Kessar. This is all joint work with Charles Eaton.
Sinead Lyle: On bases of Specht modules corresponding to 2-column partitions
In this talk, we consider Specht modules and simple modules corresponding to 2-column partitions. Decomposition numbers for Specht modules corresponding to such partitions have long been known; as have homomorphims between them. For each such partition $\lambda$, we look at certain sets of standard $\lambda$-tableaux which can be defined combinatorially in terms of paths and which naturally label a basis of the simple module $D^\lambda$. We also prove that the $q$-character of $D^\lambda$ can be described in terms of this sets. We consider the extension to 3-column partitions.
Gunter Malle: Representations of finite groups
Representation theory of finite groups was initiated by Frobenius and Burnside at the beginning of the twentieth century and then extended to the modular setting by Brauer more than 60 years ago. Still, the area abounds with intriguing fundamental open problems. In this survey talk we will review several of the famous conjectures and present recent advances and results.
Rowena Paget: Tableaux methods for plethysm
The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m. The irreducible constituents of the associated ordinary character are largely unknown; in particular they are the subject of the long-standing Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups and with plethysms of symmetric functions. In this talk I will review the plethysm problem, recall joint work with Anton and Mark Wildon and also discuss more recent work with Mark Wildon on maximal and minimal constituents of plethysms. Both rely upon tableaux combinatorics.
Shunsuke Tsuchioka: A definition of Schur regular partitions
In 1926, Schur singled out a subset of strict partitions (now called Schur regular partitions) and proved a mod 6 analog of Rogers-Ramanujan identities (or better saying, Rogers-Ramanujan partition theorem). In 1994, Bessenrodt-Morris-Olsson proved that it is a good label for 3-modular irreducible spin representations of the symmetric groups with respect to the 3-modular decomposition matrices. Furthermore, they define the characteristic 5 version of Schur regular partitions. Unfortunately, in virtue of Lascoux-Leclerc-Thibon-Ariki theory, finding a label of $p$-modular irreducible spin representations of the symmetric groups as a subset of strict partitions now seems to be abandoned. In this talk, we define a subset "$p$-Schur regular partitions" of strict partitions for any odd integer $p\geq 3$ and prove some properties indicating that this may be a good label (when $p$ is odd prime) as an application of Kashiwara crystal theory. This is a joint work with Masaki Watanabe.
James Whitley: Brauer Correspondence for Hecke algebras and the Dipper--Du conjecture
Let $H_n$ be the Iwahori Hecke algebra corresponding to the symmetric group. In this talk we consider a Brauer correspondence for the blocks of $H_n$ and classify the vertices of the blocks. We then use this classification to resolve the Dipper--Du conjecture regarding the vertices of indecomposable $H_n$-modules.