Finite Simple Groups
and Their Representations

Sunset over the UA Mall and
    Old Main. Photographer Credit: Gregor Orbino.

Programme

The programme, including the title and abstracts, can be downloaded here. Please note that all talks will take place in ENR2 S395.

Saturday Sunday
08:50 - 09:00 Introduction
09:00 - 09:50 Aschbacher 08:30 - 09:20 Guralnick
10:00 - 10:50 Geck 09:30 - 10:20 Hiß
11:00 - 11:30 Break 10:30 - 11:00 Break
11:30 - 12:20 Kleshchev 11:00 - 11:50 Breuer
12:30 - 14:30 Lunch 12:00 - 14:00 Lunch
14:30 - 15:20 Schaeffer-Fry 14:00 - 14:50 Ryba
15:30 - 16:20 Taylor 15:00 - 15:50 Navarro

Titles and Abstracts

Michael Aschbacher - Quaternion Fusion Packets

Abstract: I'll discuss a theorem characterizing the 2-fusion systems of groups of Lie type of odd characteristic. The theorem is a major step in a program to (essentially) classify the simple 2-fusion systems of component type, and to use that classification to simplify the proof of the theorem classifying the finite simple groups.


Thomas Breuer - On the Verification of ATLAS Information

Abstract: I will report on details and side-effects of the recent verification of most of the character tables in the ATLAS of Finite Groups.


Meinolf Geck - On the characters of reductive groups with a disconnected center

Abstract: There are three bases of the space of class functions on a finite group of Lie type: irreducible characters, almost characters and characteristic functions of character sheaves. We discuss various issues and open questions related to the interrelation of these three bases, especially for groups with a disconnected center.


Bob Guralnick - Simple Algebraic Groups and Stabilizers of Polynomials

Abstract: Let V be an irreducible module for the simple algebraic group G over a field k. Suppose f in k[V] is fixed by G. We show that almost always $G$ is the connected part of the stabilizer of f. In particular, we give a quick solution to a 125 year old question of Cartan to characterize E_8 and also show that in most cases simple algebraic groups can be described as stabilizers of quadratic or cubic polynomials. This is joint work with Skip Garibaldi.


Gerhard Hiß - The modular Atlas project

Abstract: The aim of this talk is to give a survey on the state of the art in the modular Atlas project, the program to compute the modular character tables of the sporadic groups. I will also present some of the methods used, in particular the MeatAxe and Condensation.


Sasha Kleshchev - RoCK blocks of symmetric groups and Hecke algebras

Abstract: We present a joint result with Anton Evseev, which describes every block of a symmetric group up to derived equivalence as a certain Turner double algebra. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. This description was conjectured by Will Turner. It relies on the work of Chuang-Kessar and Chuang-Rouquier. (RoCK=Rouquier+Chuang+Kessar). Key idea is a connection with Khovanov-Lauda-Rouquier algebras and their semicuspidal representations.


Gabriel Navarro - Local Blocks with One Simple Module

Abstract: To find the exact relationship between correspondent blocks under Brauer's First Main Theorem is one of the problems in Representation Theory. We find a characterization of when local blocks have one simple module for odd primes and principal blocks. This characterization is detectable in the character table. For p=2, we can only conjecture what is going on. This is joint work with P. H. Tiep and C. Vallejo.


Alex Ryba - The 5-modular character table of the Lyons sporadic simple group

Abstract: I will discuss ongoing computational work with Klaus Lux to find the 5-modular character table of the Lyons group Ly. We know what is undoubtedly the answer, and have a (partially completed) strategy for proving that this answer is indeed correct.


Mandi Schaeffer-Fry - On a Conjecture of G. Navarro and Self-Normalising Sylow 2-Subgroups in Type A

Abstract: G. Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalising Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of G. I will give some background on this conjecture and my reduction to showing that certain related statements hold when $G$ is quasisimple. I will then discuss recent work with Jay Taylor, in which we show that these conditions are satisfied when G/Z(G) is PSL(n,q), PSU(n,q), or a simple group of Lie type defined over a finite field of characteristic 2.


Jay Taylor - Action of automorphisms on irreducible characters of symplectic groups

Abstract: In recent years a new approach has been developed to several long standing conjectures in the representation theory of finite groups; such as the McKay conjecture. These conjectures are stated for all finite groups but the recent approach has reduced these conjectures to checking certain conditions on quasisimple finite groups (often referred to as inductive conditions). This, in theory, makes the conjecture more manageable thanks to the classification of finite simple groups. However, the downside to this is that one requires information about how automorphisms act on the irreducible characters of quasisimple finite groups. In this talk we present new results in this direction concerning the finite symplectic groups Sp(2n,q) where q is an odd prime power. Specifically we completely describe the action of the automorphism group on the ordinary irreducible characters of these groups.

Last Updated: 21/03/2017