# Finite Simple Groups

and Their Representations

## 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