Representation Theory at the Villa Denis, Oct. 2019

Tuesday, 15th of October

The notes were taken by Olivier Dudas.

Noelia Rizo Carrión [Università degli Studi di Firenze]
Title: Galois action on the principal block
Abstract:
A classical question in the representation theory of finite groups is to determine what properties of a group or its local structure can be obtained from its character table. In particular the study of the relations between the set \({\rm Irr}(G)\) of the irreducible complex characters of \(G\) and the structure of its Sylow \(p\)-subgroups has been one of the cornerstones of the character theory of finite groups. If one wants to go deeper, there is a more sophisticated version of this: looking at the irreducible characters in the principal Brauer \(p\)-block of \(G\).

In this work, we characterize finite groups \(G\) having a cyclic Sylow \(p\)-subgroup in terms of the action of a specific Galois automorphism on the principal \(p\)-block of \(G\), for \(p=2,3\). We conjecture an analog statement for blocks with arbitrary defect group and we prove that this general statement would follow from the blockwise Galois–McKay conjecture.

Olivier Dudas [Université Paris Diderot]
Title: Unitriangular shape of decomposition matrices of unipotent blocks
Abstract:
[With O. Brunat and J. Taylor] In this talk we will be interested in the decomposition matrices of finite reductive groups (e.g. GL(n,q) or Sp(2n,q)). Such a matrix encodes how ordinary representations decompose when they are reduced to a field with positive characteristic \(\ell\). I will explain how one can use Kawanaka's recipe to construct projective modules with few irreducible constituents. When \(\ell\) is not too small, this forces the decomposition matrix of any unipotent block to have a unitriangular shape, as predicted by Geck. If time permits I will discuss some generalisation to the case of bad characteristic.

Emily Norton [TU Kaiserslautern]
Title: Harish-Chandra series of unipotent representations of finite unitary groups
Abstract:
A basic problem in the modular representation theory of finite groups of Lie type is to describe the Harish-Chandra series of simple modules in unipotent blocks. In the case of finite unitary groups when the characteristic of the ground field is a unitary prime, major progress on this problem was made in the last five years by Dudas, Gerber, Hiss, Jacon, Varagnolo, and Vasserot. Their works show that a certain graph structure on level 2 Fock spaces, the "sl_e hat crystal," describes part of the Harish-Chandra series of a simple module in a unipotent block. This is identical to the partial solution to the same problem for rational Cherednik algebras of type B Weyl groups. In this talk I will describe how considering a second graph structure on level 2 Fock spaces results in a complete description of the Harish-Chandra series of simple modules in unipotent blocks of finite unitary groups, provided the characteristic of the ground field is large enough. This second graph structure is the same as the one which provides the rest of the solution to the same problem for rational Cherednik algebras of type B.

Lucas Ruhstorfer [TU Kaiserslautern]
Title: Jordan decomposition for the Alperin-McKay conjecture
Abstract:
In recent years, many of the famous global-local conjectures in the representation theory of finite groups have been reduced to the verification of certain stronger conditions on the characters of finite quasi-simple groups. It became apparent that checking these conditions requires a deep understanding of the action of \(\mathrm{Aut}(G)\) on the characters of a finite simple group \(G\) of Lie type.

On the other hand, the Morita equivalence by Bonnafé--Dat--Rouquier has become an indispensable tool to study the representation theory of groups of Lie type. In this talk, we will discuss the interplay of this Morita equivalence with group automorphisms. We will then show how this can be applied in the context of the Alperin--McKay conjecture.