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Paul Balmer: The geometry of permutation modules

5th October, 2023

In joint work with Martin Gallauer, we study the tensor-triangular geometry of the derived category of permutation modules over a finite group, and more generally over profinite groups. Martin and I have already spoken on this topic in various venues. So I’ll try to comment on aspects that were not highlighted so far, like the construction of the modular fixed points (or Brauer quotients), the Koszul objects and the reduction to elementary abelian groups. If time permits, I’ll say a few words about the profinite case, which is still partially work-in-progress.

Soichi Okada: Intermediate symplectic characters and enumeration of shifted plane partitions

14th March, 2023

The intermediate symplectic characters, introduced by R. Proctor, interpolate between Schur functions and symplectic characters. They arise as the characters of indecomposable representations of the intermediate symplectic group, which is defined as the group of linear transformations fixing a (not necessarily non-degenerate) skew-symmetric bilinear form. In this talk, we present Jacobi-Trudi-type determinant formulas and bialternant formulas for intermediate symplectic characters. By using the bialternant formula, we can derive factorization formulas for sums of intermediate symplectic characters, which allow us to give a proof and variations of Hopkins' conjecture on the number of shifted plane partitions of double-staircase shape with bounded entries.

Pavel Turek: On stable modular plethysms of the natural module of SL2(𝔽p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(𝔽p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(𝔽p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(𝔽p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

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.

Giada Volpato: On the restriction of a character of Sn to a Sylow p-subgroup

15th November, 2022

The relevance of the McKay Conjecture in the representation theory of finite groups has led to investigate how irreducible characters decompose when restricted to Sylow p-subgroups. In this talk we will focus on the symmetric groups. Since the linear constituents of the restriction to a Sylow p-subgroup has been studied a lot by E. Giannelli and S. Law, we will concentrate on constituents of higher degree. In particular, we will describe the set of the irreducible characters which allow a constituent of a fixed degree, separating the cases of p being odd and p=2.