Seminars in Representations of Finite Groups

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Carlos André: Supercharacters of adjoint groups of radical rings and related subgroups

25th November, 2024

Describing the conjugacy classes and/or irreducible characters of the unitriangular group over a finite field is known to be an impossibly difficult problem. Superclasses and supercharacters have been introduced (under the names of "basic varieties" and "basic characters") as an attempt to approximate conjugacy classes and irreducible characters using a cruder version of Kirillov's method of coadjoint orbits. In the past thirty years, these notions have been recognized in several areas (seemingly unrelated to representation theory): exponential sums in number theory, random walks in probability and statistics, association schemes in algebraic combinatorics... In this talk, we will describe and illustrate the main ideas and recent developments of the standard supercharacter theory of adjoint groups of radical rings. We will explore the close relation to Schur rings, and extend a well-known factorization of supercharacters of unitriangular groups which explains the alternative definition as basic characters.

Dave Benson: The nucleus and the singularity category of cochains on the classifying space

30th May, 2024

The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.

Charlotte Chan: Generic character sheaves on parahoric subgroups

27th May, 2024

Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the 'generic' part of this desired theory. In the simplest non-trivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne-Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands programme.

Jon Carlson: Endotrivial modules for finite groups of Lie type

27th May, 2024

Suppose that G is a finite group and that k is a field of characteristic p > 0. A kG-module M is an endotrivial module if Homk(M,M) ≅ M∗ ⊗ M ≅ k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension 1, but the exceptions are notable.

Eric Marberg: From Klyachko models to perfect models

5th December, 2023

In this talk a "model" of a finite group or semisimple algebra means a representation containing a unique irreducible subrepresentation from each isomorphism class. In the 1980s Klyachko identified an elegant model for the general linear group over a finite field with q elements. There is an informal sense in which taking the q→1 limit of Klyachko's construction gives a model for the symmetric group, which can be extended to its Iwahori-Hecke algebra. The resulting Hecke algebra representation is a special case of a "perfect model", which is a more flexible construction that can be considered for any finite Coxeter group. In this talk, I will classify exactly which Coxeter groups have perfect models, and discuss some notable features of this classification. For example, each perfect model gives rise to a pair of related W-graphs, which are dual in types B and D but not in type A. Various interesting questions about these W-graphs remain open.

Sam Miller: Endotrivial complexes

20th November, 2023

Let G be a finite group, p a prime, and k a field of characteristic p. In this talk, we will introduce the notion of an endotrivial complex of p-permutation kG-modules, and the corresponding group of endotrivial complexes. Such complexes induce splendid Rickard autoequivalences of the group algebra kG. These complexes can be determined up to homotopy equivalence by integral invariants arising from the Brauer construction and a 1-dimensional representation, which proves that the group of endotrivial complexes is finitely generated. We will discuss some of the results of our investigations, including explicitly determining the group of endotrivial complexes for certain groups, investigating the image of the group in the trivial source ring, and restriction to Sylow p-subgroups. If time permits, we will briefly discuss ongoing work which defines the notion of a relative endotrivial complex, which extends Lassueur's doctoral thesis.

Benjamin Sambale: Groups of p-central type

10th October, 2023

A finite group G with centre Z is of central type if there exists an irreducible character χ such that χ(1)2=|G:Z|. Howlett–Isaacs have shown that such groups are soluble. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p≠5. I have shown that there are no exceptions for p=5. Moreover, I give some applications to p-blocks with a unique Brauer character.

Geordie Williamson: A Panoramic View of Modular Representation Theory

6th October, 2023

I will try to give a glimpse of exciting developments in representation theory over the last two decades. A central focus will be on the representations of symmetric groups over the complex numbers and fields of positive characteristic. Over the complex numbers our understanding is very good, however the case of positive characteristic fields has turned out to be more complicated than (I suspect) the pioneers would have ever imagined. Remarkably, there appears to be a way forward which combines ideas which emerged in the Langlands programme with techniques from mod p algebraic topology (Smith theory).