- Bernhard Böhmler [TU kaiserslautern]
Titel: On the computation of trivial source character tables using
computer algebra
Abstract:
Let $G$ be a finite group and $k$ be a large enough field of
characteristic $p$ dividing $|G|$.
The Brauer correspondence establishes a bijection between the
isomorphism classes of
indecomposable $kG$-modules with trivial source and vertex $P$ and the
isomorphism classes of non-zero projective indecomposable
$k[N_G(P)/P]$-modules.
In this talk, we explain how to explicitly calculate matrix
representations of projective indecomposable modules using the open
source computer algebra system GAP.
Moreover, we comment on how these results can be applied to the
computation of the whole trivial source character table of $kG$.
- Thomas Breuer [RWTH Aachen]
Titel: Finite Groups can be generated by a \(\pi\)-subgroup and a \(\pi'\)-subgroup
Abstract:
Given a set of primes \(\pi\), any finite group can be generated by a \(\pi\)-subgroup and a \(\pi'\)-subgroup. The proof is based on the Classification of Finite Simple Groups. This is joint work with Robert M. Guralnick.
- Eirini Chavli [Universität Stuttgart]
Title: Nakayama algebras via combinatorics
Abstract:
Nakayama algebras appear in modular representation theory of groups and they are defined as a quotient of the algebra of upper triangular matrices. In this talk we will explain the connection of these algebras with two combinatorial objects: the 321-avoiding permutations and the Dyck paths. We will use these objects to explain the representation theory and homological algebra of Nakayama algebras (joint work with René Marczinzik).
- Meinolf Geck [Universität Stuttgart]
Title: Ree's formula and character sheaves
Abstract:
Ree's formula expresses the values of the principal series characters of a finite group
of Lie type in terms of characters of Hecke algebras and intersections of conjugacy classes
with Bruhat cells. We give some recent examples on how this can be used to obtain subtle geometric information about character sheaves.
- Thomas Gerber [EPFL, Ecublens]
Titel: Looking for cores
Abstract:
It is well-known that block theory for the symmetric group is controlled by the combinatorics of core partitions. This has led to various generalisations of the notion of core, controlling block theory for other structures such as finite groups of classical type or Hecke algebras.
In 1996, Granville and Ono proved the existence of an \(e\)-core of every size as soon as \(e\geq 4\). This enabled to complete the celebrated proof that every finite simple group has a defect zero \(p\)-block for \(p\geq 5\).
In this talk, we are interested in analogous results for the different generalisations of cores. We will report on work in progress with Emily Norton and with Nathan Chapelier.
- Gerhard Hiß [RWTH Aachen]
Titel: On the source algebra equivalence class of blocks with cyclic defect groups
Abstract:
This is joint work with Caroline Lassueur. The source algebra equivalence class of a p-block B with cyclic defect group D in a finite group G is determined by the embedded Brauer tree of B, a type function on the vertices of this Brauer tree and an endo-permutation module W(B) of D. We show how W(B) can be detected from the character table of G. Given a finite cyclic p-group D, we also work out those endo-permutation modules of D that occur as W(B) for p-blocks B with defect group D in quasisimple groups.
- Birte Johansson [TU Kaiserslautern]
Titel: On the verification of the inductive McKay-Navarro condition
Abstract:
Navarro's refinement of the McKay conjecture claims that the bijection from the McKay conjecture can be chosen such that it is equivariant under certain Galois automorphisms. In 2019, Navarro, Späth, and Vallejo reduced the McKay-Navarro conjecture to a problem about simple groups, resulting in an inductive condition. We introduce the inductive McKay-Navarro condition and present recent progress on its verification for some finite groups of Lie type.
- Caroline Lassueur [TU Kaiserslautern]
Titel: Lifting endo-permutation modules in characteristic 2
Abstract:
In this talk, we explain a result about the lifting of endo-permutation modules from characteristic two to characteristic zero, and emphasise the differences with the odd characteristic case. This fills a small gap left open by Puig, and allows to prove that further classes of modules, such as the endo-p-permutation or the Brauer-friendly modules, are classes of liftable modules. The talk will also serve as an introduction to the objects considered in the following talk by G. Hiss.
- Ivan Marin [Université de Picardie, Jules Verne]
Title: Reflection subgroups and their normalizers
Abstract:
I will review several recent results about reflection subgroups of real or complex reflection groups and related algebras. These algebras are associated either to the normalizer of such a reflection subgroup, or to the whole collection of them -- it turns out however that these two kind of algebras are closely related.