Tag - Modular representation theory

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.

Vera Serganova: Sylow theorems and support varieties for supergroups

30th May, 2024

We discuss support variety theory for quasireductive algebraic supergroups, i.e. supergroups with reductive even part over complex numbers. The corresponding categories of representations are Frobenius and share many properties of representations of finite groups in positive characteristic. It is desirable to describe Balmer spectrum of related triangulated symmetric monoidal categories. Our approach involves so called homological odd elements and certain tensor functors associated to them. On the way we encounter analogues of p-groups and Sylow subgroups for supergroups. We prove projectivity detection for our support theory and present other related results. We also explore connections with homological support theory developed by B. Boe, J. Kujawa and D. Nakano.

Eoghan McDowell: Spin representations of the symmetric group which reduce modulo 2 to Specht modules

16th April, 2024

When do two ordinary irreducible representations of a group have the same p-modular reduction? In this talk I will address this question for the double cover of the symmetric group, and more generally give a necessary and sufficient condition for a spin representation of the symmetric group to reduce modulo 2 to a multiple of a Specht module (in the sense of Brauer characters or in the Grothendieck group). I will explain some of the techniques used in the proof, including describing a function which swaps adjacent runners in an abacus display for the labelling partition of a character.

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).

Vera Serganova: Invariant integration on homogeneous superspaces

1st September, 2023

We use Berezin integral in the category of CS-manifolds to construct an invariant integral for the ring of regular functions on a homogeneous affine supervariety G/K. This construction has several applications in representation theory of G. We will explain how it is used in the proof of projectivity detection for support varieties and for description of stable categories for defect 1 supergroups. We also see how this integral can be used to generalize some classical statements from modular representation theory of finite groups to supergroups in characteristic zero.

Matyas Domokos: Improvements of the Noether bound for polynomial invariants of finite groups

24th November, 2022

Given a field and a finite group G, the Noether number of G is defined as the minimal positive integer d such that for any finite dimensional G-module V, the algebra of G-invariant polynomial functions on V is generated by elements of degree at most d. In the talk we shall survey results (obtained mostly together with Kálmán Cziszter) on the Noether number of various finite groups.

Haralampos Geranios: On self-extensions of irreducible modules for symmetric groups

11th October, 2022

We work in the context of the modular representation theory of the symmetric groups. A long-standing conjecture, from the late 80s, suggests that there are no (non-trivial) self-extensions of irreducible modules over fields of odd characteristic. In this talk we will highlight several new positive results on this conjecture.