Agenda

Given a finite group G, S a Sylow p-subgroup, and k a field of characteristic p, the induced module Ind𝑆𝐺(𝑘) is known as the Sylow permutation kG-module. This talk  will  have as focus  certain summands of the Sylow permutation module arising from source algebras of the blocks of kG and the corresponding endomorphism algebras.  I will present general structural results as well as an application to a question of Persi Diaconis on the self-injectivity of the endomorphism algebra of the Sylow permutation module of a finite symmetric group.  

This is joint work with  Markus Linckelmann.

One of the central themes in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and those of its local subgroups. In particular, Sylow branching coefficients describe how an irreducible character of G decomposes upon restriction to a Sylow subgroup of G. 

In this talk, we will discuss some of their connections to broader group-theoretic questions, before presenting new results in the case of symmetric groups in joint work with E. Giannelli.

The arithmetic structure of character degrees of finite groups is a rich and still not well understood topic. The recently proved Height Zero Conjecture provides a strong connection to the structure of defect groups. A generalisation was proposed by Eaton and Moretó. We report on recent progress towards their conjecture. 

This is joint work with A. Moretó, N. Rizo, A. Schaeffer Fry.

Let G be a finite group, p be a prime dividing the order of G, P  a Sylow p-subgroup of G. The Itô-Michler theorem states that p does not divide the degree of any irreducible complex character of G if and only if P is abelian and normal in G. It turns out that the abelian and normal parts of this statement can be separated and characterized in character-theoretic terms. Recently, G. Malle, G. Navarro, and P. H. Tiep have proposed a new way of determining the normality of P  in G in terms of Brauer characters, different in nature from the previously-known characterizations. In this talk, we report on the progress on this conjecture. 

This is joint work with Z. Feng and D. Rossi.

When do two ordinary irreducible representations of a group have the same p-modular reduction? Equivalently, when do two ordinary irreducible characters have the same values on p-regular classes? I will present an (almost complete) answer to this question for the double cover of the symmetric group. I will briefly discuss some of the techniques used in the proofs, which include applications of central characters and of induction and restriction functors.

 Part of this work is joint with Matt Fayers.

In the 28 papers I coauthored with I. M. Isaacs between 1991 and 2024, several questions remain open. In this talk, I will survey some of these problems as a tribute to I. M. Isaacs.

One of the central topics in representation theory of finite groups is the study of local-global conjectures. For instance, the recently proven McKay conjecture provides a, yet to be fully understood, relation between the values of the irreducible characters of a finite group at the identity element and those of the irreducible characters of its Sylow normalizer.

Very recently, Alexander Moretó has announced a series of conjectures that, to some extent, generalize the classical local-global ones.  These new problems involve the so-called picky elements (elements that belong to a unique Sylow subgroup) as well as their subnormalizers. The aim of this talk is to introduce these new conjectures and to present some related results.

This is joint work with A. Moretó.

In his celebrated papers Counting characters in blocks I, II, E. C. Dade proposed a conjecture giving a precise formula for counting the number of characters of any given p-defect in a Brauer p-block of a finite group, where p is a fixed prime number. For finite groups of Lie type defined over finite fields of characteristic different from p, it is a remarkable fact that Brauer's theory of p-blocks aligns with Lusztig's theory. This leads to the study of so-called Brauer-Lusztig blocks: intersections of Brauer blocks with Lusztig series. In this talk, I will explain the relationship between Brauer's and Lusztig's theory and introduce a version of Dade's Conjecture for Brauer-Lusztig blocks. This work is motivated by the ultimate goal of studying Dade's Conjecture through the use of the Jordan decomposition of characters.

There are many known results on groups with few rational conjugacy classes. For example, it is known that the groups with a single rational conjugacy class are exactly the groups of odd order; these are, in turn, exactly the groups with a single rational irreducible character. In this talk, we will look at groups on the opposite extreme: those where almost all conjugacy classes are rational. In particular, we obtain a condition to guarantee that the number of rational irreducible characters and that of rational conjugacy classes are the same for those groups. In the process, we also find results that seem to indicate a connection between groups with many rational conjugacy classes and those with few  rational irreducible characters.

 If 𝐺(𝑞) is a finite reductive group, where 𝑞 is a prime power, then Lusztig has defined a set of complex irreducible characters UCh(𝐺(𝑞)) called the unipotent characters of the group. Somewhat remarkably these characters are parameterised independently of 𝑞.

The Lie algebra 𝔤 of the underlying algebraic group 𝐺 admits a ℤ-form 𝔤₀ ⊆ 𝔤 whose elements make sense for any 𝑞. In this talk I will outline joint work with Adam Thomas (Warwick) where we find nilpotent elements of 𝔤₀ which behave very well when reduced modulo any prime. Moreover, I will explain the relationship with Lusztig's classification of unipotent characters.

The irreducible representations of the symmetric group in characteristic p are indexed by p-regular partitions. There is a bijection from p-regular partitions to p-regular partitions called the Mullineux map, which arises from tensoring irreducible representations with the signature representation. In 1979, Mullineux conjectured a combinatorial description of the Mullineux map, which awaited its proof for almost 20 years. In the talk, I introduce a new abacus algorithm for computing the Mullineux map based on an earlier algorithm by Xu. This novel algorithm can be used to prove a new surprising behaviour of the Mullineux map applied to partitions with arms divisible by a fixed integer d coprime to p. I present this result as well as some of the rich combinatorial properties of these partitions, which may be of independent interest.