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Thursday, 22nd of February, Room F13.11
- 13:00 - 14:00: coffee in F13.12
- 14:00 - 14:30: Steffen Kionke
- 14:40 - 15:20: Margherita Piccolo
- 15:20 - 15:50: coffee break
- 15:50 - 16:40: Arnaud Eteve
- 17:10 - 17:40: Gunter Malle
- 19:00 - : Dinner, Restaurant: Goldalm [Herzogstraße 26,]
Friday, 23rd of February, Room F13.11
- 09:00 - 09:50: Noelia Rizo
- 10:00 - 10:30: Sofia Brenner
- 10:30 - 11:00: coffee break
- 11:00 - 11:50: Damiano Rossi
- 12:00 - 12:30: Meinolf Geck
TALKS / ABSTRACTS
- Sofia Brenner [TU Darmstadt]
Title: A case of the modular isomorphism problem for odd primes
Abstract: The modular isomorphism problem is the famous question whether two non-isomorphic p-groups G and H can have isomorphic group algebras over the field with p elements for a prime number p. Open for over 50 years, a counterexample for p = 2 was found by García-Lucas, Margolis and del Río in 2021. In this talk, I will discuss a case of this problem for odd p, which is inspired by the counterexample for p = 2. This is joint work with Diego García-Lucas.
- Arnaud Eteve [MPI Bonn]
Title: On the endomorphism of the Gelfand-Graev representation
Abstract: Let G be a reductive group with connected center over k, the algebraic closure of a finite field of characteristic p > 0, let F be Frobenius endomorphism of G and let U be the unipotent radical of an F-stable Borel. Let ψ be a generic character of UF with values in \bar Zℓ* and let Γ ψ be the corresponding Gelfand-Graev representation, obtained by inducing ψ to GF . Assuming ℓ to be different from p and a good prime for G, a theorem of Shotton-Li gives a description of the endomorphism algebra EndGF(Γ ψ) in terms of the Langlands dual group of G. In our thesis, we reformulated some key constructions of Deligne-Lusztig theory using some categorical methods known as categorical traces. In this talk, we will illustrate these technics and explain how to use them to produce a new proof of the theorem of Shotton-Li.
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Meinolf Geck [Universität Stuttgart]
Title: Canonical structure constants for simple Lie algebras
Abstract: Let L be a complex simple Lie algebra with root system R. By a famous result of Chevalley, one can choose root elements er in L (r in R) such that the corresponding structure constants are integers. Here, the er are only unique up to signs. But, by Lusztig's theory of canonical bases, it is possible to single out a canonical choice for all er. Question: Can one give explicit formulae for the corresponding structure constants? The answer is YES, and due to Alexander Lang in his recent Master thesis at Stuttgart. - Steffen Kionke [FU Hagen]
Title: Representation zeta functions à la Weil
Abstract: The Weil representation zeta function of a group G is a generating function counting the absolutely irreducible representations of G over all finite fields. It is reminiscent of the Hasse-Weil zeta function of algebraic varieties and converges for the large class of UBERG groups. We give a short introduction and present some examples. We indicate why the value at k encodes the probability that k random elements generate the completed group ring and briefly discuss finite and virtually abelian groups.
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Gunter Malle [RPTU Kaiserslautern]
Title: Rationality of extended unipotent characters
Abstract: Recent research around the McKay-Navarro conjecture on Galois actions on characters of finite groups requires knowledge on the character fields for finite nearly simple groups. We investigate the rationality properties of unipotent characters of finite reductive groups extended by a graph automorphism using realisations of characters in ℓ-adic cohomology groups of Deligne--Lusztig varieties as well as block theoretic considerations. This is joint work with Olivier Dudas. - Margherita Piccolo [HHU Düsseldorf]
Title: Representation growth of semisimple and quasisemisimple profinite groups
Abstract: A profinite group is called semisimple (resp. quasisemisimple) if it is the Cartesian product of finite simple groups (resp. quasisimple groups). The representation growth of such groups can be studied looking at the distribution of irreducible representations by means of a zeta function, that is a Dirichlet generating function. Under certain restrictions, the representation growth is polynomial with a wide range of growth.
Moreover, given a profinite group, it is generally a difficult question to determine if it is, in fact, isomorphic to a profinite completion of an abstract group. In this talk, I discuss a result of Kassabov and Nikolov which provides a criteria for a semisimple profinite group to be a profinite completion. Based on this, I report on my work which is aimed at constructing semisimple and quasisemisimple profinite groups with specified polynomial representation growth. Moreover, the semisimple profinite groups arise as profinite completions of abstract groups (with the same representation growth).
- Noelia Rizo [Universitat of Valencia]
Title: A Brauer–Galois Height Zero Conjecture
Abstract: In 1963, Brauer published a list of 43 problems that he saw as the main questions to be answered in group theory and character theory. This list has lead the research in the field of representation theory of finite groups since then. Many important problems in our area are collected in this list, such as the very recently solved Brauer’s Height Zero Conjecture.In [MN] Malle and Navarro obtained a Galois strengthening of Brauer’s height zero conjecture for principal 2-blocks, considering a particular Galois automorphism of order 2. In this talk we will discuss a version for odd primes p of this result presented [G. Malle, A. Moreto, N. Rizo, A. A. Schaeffer Fry]. This is a joint work with G. Malle, A. Moreto and M. Schaeffer Fry.
- Damiano Rossi [University of Loughborough]
Title: The homotopy type of the Brown complex of a finite reductive group
Abstract: The Brown complex is the simplicial complex associated with the poset of non-trivial p-subgroups of a finite group: its simplices can be thought of as chains of p-subgroups. The topology of the Brown complex can be seen to control the algebraic p-local structure of the group. For a finite reductive group defined over a field of characteristic p, Quillen proved that the Brown complex is homotopy equivalent to the Tits building. We will show that in the case of a finite reductive group defined over a field of characteristic different from p, the homotopy type of the Brown complex can be described in terms of the generic Sylow theory introduced by Broué and Malle.
