10th meeting: Groups & Representations

University of Birmingham, Wednesday January 25th 2017

All lectures will take place Lecture Room B of the Watson Building.

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).

14:00-15:00 Carolina Vallejo Rodríguez (ICMAT) Detecting local properties in the character table

Let G be a finite group and let p be a prime number. In this talk, we discuss local properties of G that can be read off from its character table. More precisely, we characterize globally when the principal block of the normalizer of a Sylow p-subgroup has one simple module for p odd. We also talk about the p=2 case of this problem, which remains open. This is joint work with G. Navarro and P. H. Tiep.

15:00-16:00 Eugenio Gianelli (Cambridge) Characters of odd degree of symmetric groups

Let G be a finite group and let P be a Sylow p-subgroup of G.

Denote by Irrp(G) the set consisting of all irreducible characters of G of degree coprime to p.

The McKay Conjecture asserts that |Irrp(G)|=|Irrp(NG(P))|.

Sometimes, we do not only have the above equality, but it is also possible to determine explicit natural bijections (McKay bijections) between Irrp(G) and Irrp(NG(P)).

In the first part of this talk I will describe the construction of McKay bijections for symmetric groups at the prime p=2.

In the second part of the talk I will present a recent joint work with Kleshchev, Navarro and Tiep, concerning the construction of natural bijections between IIrrp(G) and Irrp(H) for various classes of finite groups G and corresponding subgroups H of odd index. This includes the case G=Sn and H any maximal subgroup of odd index in Sn, as well as the construction of McKay bijections for solvable and general linear groups.

16:00-16:30 Break

16:30-17:30 Geoff Robinson (Aberdeen/Lancaster) On a subgroup introduced by J.Grodal

(report on ongoing joint work with J. Grodal).  We will discuss the structure of the (normal) subgroup of a finite group G generated by the elements whose centraliser has order divisible by the prime p.  This leads quickly to a study of an interesting generalization of Frobenius complements.

The abelianization of the associated quotient group plays a role in J. Grodal’s work on endotrivial modules.