TALKS / ABSTRACTS (received so far)

• Olivier Dudas [Aix-Marseille Université]
  Titel: Row and column removal rules for finite unitary groups
  Abstract: James gave in the 80's a construction of the unipotent representations of the finite general linear groups GL(n,q), very similar to the construction of Young, Specht and simple modules for the symmetric groups. This construction was then used to show a combinatorial property for the decomposition numbers (which can be computed as the multiplicity of a simple module in a Specht module). It uses two ingredients:
  - Harish-Chandra induction/restriction from Levi subgroups of GL(n,q);
  - Linear character of the group of unitriangular matrices (yielding Gelfand-Graev representations).
  In a work in progress with Pengcheng Li, we propose a generalisation of this construction to finite general unitary groups, replacing the Harish-Chandra induction by the Howe correspondence, and the Gelfand-Graev representations by their generalisation to other unipotent classes.
  
  
• Arnaud Eteve [MPIM Bonn]
  Titel: Tilting representations
  Abstract: Let \(G\) be a finite reductive group of defining characteristic \(p\) and \(\ell \neq p\) a prime. The goal of this talk is to discuss a work in progress on a generating collection of integral projective representations of \(G\) which we call tilting representations. Their definition is motivated by the theory of tilting objects in Hecke categories and connexions between the representation theory of \(G\) and Hecke categories. Hecke categories are equipped with a Koszul duality exchanging simple and tilting objects, in our setting we discuss the relation between these tilting representations and the Alvis-Curtis dual of the representations coming from the intersection cohomology of Deligne-Lusztig varieties.
  
  
• Sonia Petschick [Bergische Universität Wuppertal]
  Title: Steps towards the inductive Galois-McKay conditions for type A
  Abstract: Navarro's refinement of the McKay conjecture suggests that the bijection from the original conjecture can be made equivariant under specific Galois automorphisms. Like the McKay conjecture, this refinement was reduced to inductive statements for quasi-simple groups by Navarro, Späth, and Vallejo in 2019. In this talk we will explore which tools and techniques can be applied to establishing the inductive conditions of the Galois refinement for simple groups of type A. In particular we focus on the challenge of achieving the equivariance property.
  
  
• Damiano Rossi [RPTU KL-LD]
  Title: An inductive version of Alperin's lower bound on Brauer characters
  Abstract: In this talk, I will recall the statement of Alperin's Weight Conjecture and some of its consequences. I will then present some old and new reduction theorems for these statements to question about finite simple groups and show how they can be used to obtain new unexpected results. In particular, I will show how a new reduction theorem for Alperin's bound on Brauer characters can be used to characterise groups with a normal Sylow 2-subgroup in terms of Brauer characters. The latter statement builds on some recent results (still work in progress) of Martìnez and Schaeffer-Fry.
  
  
• Lucas Ruhstorfer [Bergische Universität Wuppertal]
  Title: The Alperin-McKay conjecture and blocks of maximal defect
  Abstract: For blocks of maximal defect the Alperin-McKay conjecture can be seen as a blockwise refinement of the recently proved McKay conjecture. I will explain what additional block-theoretic problems arise in this situation and report on recent progress on this question. This is joint work with Britta Späth.
  
  
• Edoardo Salati [Technische Universität Dresden]
  Title: Fusion systems and localities with a large \(p\)-subgroup
  Abstract: In the framework of a project initiated by Meierfrankenfeld, Stellmacher and Stroth, whose aim consists in describing all finite simple groups of local characteristic \(p\), an additional property shared by most of the generic examples is detected and successfully employed: namely, possessing a large \(p\)-subgroup. Indeed, a milestone of the project is the Local Structure Theorem for finite groups with a large \(p\)-subgroup (LST).
  Core of the talk will be recent advancements related to the development of an analogous result for structures that function as models of the \(p\)-local information of a finite group, namely fusion systems and localities. After setting the proper initial stage, our approach allowed to use the known proof of the LST as a blueprint for many of the appearing cases; we will present the results by comparison with the LST, showing where our approach has been, and can in future be, successful and where it encounters obstructions intrinsic in the different nature of the objects we focus on.
  This project was the main topic of my doctoral dissertation under the supervision of Ellen Henke.
  
  
• Carolina Vallejo Rodríguez [Università degli studi di Firenze]
  Title: Characters and Sylow abelianization
  Abstract: In this talk, I will discuss how the character table of a finite group \(G\) can be used to determine whether the Sylow \(p\)-subgroups of \(G\) have a small abelianization. This is based in joint work with Eugenio Giannelli, Noelia Rizo and Mandi Schaeffer Fry.