Abstract:  Believe it or not, there are simple questions about the Sylow-p double cosets of the symmetric group SN  that we don't know how to answer. In joint work with Eugenio Giannelli, Bob Guralnick, Stacy Law, Gabriel Navarro and Hunter Spink we show that 'most double cosets are as large as possible' (so if N=pk, of size p2k) and the estimates are sharp enough to determine the asymptotics of the number of double cosets. We don't know lots of things: how to 'name' the double cosets? Is the associated Hecke algebra 'Frobenius' over fields of characteristic p?

Abstract:  Let G be a finite group, let p be a prime and let P be a Sylow p-subgroup of G. Motivated by ideas, conjectures, suspects, intuitions, GAP calculations, emails and messages of Gabriel Navarro, over the last few years I have been studying the decomposition into irreducible constituents of the restriction to P of irreducible characters of G. The multiplicities describing this restriction process are known as Sylow Branching Coefficients (SBCs). In this talk I will focus on symmetric groups. I will first present some new results concerning zeros of irreducible characters on elements of prime power order and then conclude by explaining how SBCs can be computed by inspecting some of the combinatorial properties of the trees and forests naturally associated to the irreducible characters of a Sylow p subgroup of Sn.

This talk is based on recent joint works with Eoghan McDowell and Stacey Law.

Abstract: We report on the determination of picky p-elements in groups of Lie type, and generalisations. These ongoing investigations are prompted by a question of A. Moretó and N. Rizo.

Abstract: I will talk about Gabriel’s influence on my work. 

Abstract: Let p be a prime and let G be a finite group, R be a suitable complete discrete valuation ring. The conjugation module for the finite group G is RG with action induced by the conjugation of G on itself. The truncated conjugation  module, which we denote by Tc,p(G) is a projective module that affords a character which vanishes on all p-singular elements of G, and whose character agrees with that of the conjugation module on all p-regular elements.This module occurs naturally in connection with the Knörr-Robinson treatment of Alperin's Weight Conjecture. We discuss two decompositions on Tc,p(G), one in terms of simple 𝔽G-modules and their projective covers (where 𝔽= R/J(R)), and one in terms of the conjugation actions of G on each of its p’-sections (leading to module actions similarly truncated to p-regular elements).

This leads to character-theoretic questions which may be of independent interest, and we consider consequences and applications in cases these questions have a positive answer.

Abstract: I present some lesser-known open problems on groups, characters and blocks.

Abstract: In recent joint work with Gabriel Navarro, Gunter Malle, and Pham Huu Tiep, we completed the proof of Brauer’s long-standing Height Zero Conjecture (BHZ), one of the first "local-global" conjectures in the representation theory of finite groups. This now-theorem says that every character in a p-block of a finite group has height zero if and only if the corresponding defect groups are abelian. The goal of this talk is to discuss several extensions of the BHZ, including the Eaton-Moretó conjecture, a Galois version, and a normal version. This will include joint work with Gunter Malle, Alex Moretó, and Noelia Rizo, along with various combinations of the four of us.

Abstract: For blocks of maximal defect the Alperin-McKay conjecture can be seen as a blockwise refinement of the McKay conjecture. Some of the techniques and results used in proving the McKay conjecture help in the context, other need some refinement. I will explain what additional block-theoretic problems arise in this situation and how they can be solved using Dade’s ramification groups. This is joint work with Lucas Ruhstorfer.

Abstract:  The generalized Gluck--Wolf theorem states that, if for a finite group G with a normal subgroup N, a fixed irreducible λ of N, and a fixed prime p, p is coprime to the ratios χ(1)/λ(1) for all irreducible characters χ of G lying above λ, then G/N has abelian Sylow p-subgroups.  This result constituted a key step towards the recent proof of Brauer's Height Zero Conjecture. We will discuss a further extension of this theorem to sets π of primes, obtained in joint work with M. Liebeck, G. Navarro, and C. Praeger.

Abstract: Associated with characters or blocks of finite groups are a number of invariants. The calculation of these invariants in as straightforward a method as possible is a desirable goal. Some of these invariants are associated to versions of well known local global conjectures in representation theory of finite groups and a simple way to calculate the invariant might help in the eventual discovery of a proof. In this talk, we will briefly discuss some of these conjectures and some of the results obtained about them. We will then touch on the questions of practical computation of certain invariants, as well as on the existence of a couple of useful uniquely defined linear characters associated to p-blocks with cyclic defect group.