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).
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.
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.
(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.