TALKS / ABSTRACTS

• Magdalena Boos [Ruhr-Universität Bochum]
  Titel: From type A to all classical Lie types: Let's discuss Symmetric Representation Theory
  Abstract: When considering representation theory of quivers, one might be disappointed that the theory only deals with type A setups, i.e. general linear groups and their Lie algebras. The notion of a symmetric quiver was first introduced by Derksen and Weyman in 2002 and allows considering classical settings in types B, C and D. We give an introduction to symmetric quiver representations, motivate our interest in the theory and show first results. This is joint work with G. Cerulli Irelli.
  
  
• Thorsten Heidersdorf [Universität Bonn]
  Titel: Deligne categories and complex representations of the finite linear group
  Abstract: Deligne defined symmetric monoidal categories \(Rep(S_t)\), \(Rep(O_t)\) and \(Rep(GL_t)\) (\(t \in \mathbb{C}\) interpolating the finite dimensional representations of \(S_n\), \(GL_n\) and $O_n$ over a field of characteristic zero. Variants and generalizations of these categories have appeared throughout the literature (e.g. by Turaev, Etingof, Knop, Flake-Maassen, Rui-Song,...).
  
  In this talk I will give a survey and then focus on the case of complex representations of the finite linear group \(GL_n(\mathbb{F}_q)\) (previously studied by Knop and Harman-Snowden). I will explain why the corresponding Deligne category is the universal symmetric monoidal category with an \(\mathbb{F}_q\)-linear Frobenius space. Then I will explain how to embed it into an abelian tensor category (its abelian envelope) in a universal way and how these categories relate to representations of \(GL_{\infty}(\mathbb{F}_q)\).
  
  This is joint work with Inna Entova-Aizenbud (Ben-Gurion).
  
  
• Nicolas Jacon [Université de Reims Champagne-Ardenne]
  Titel: Cores and weights for Ariki-Koike algebras
  Abstract: The weight and the core of a partition are two important notions in the study of the blocks of the symmetric group and its Hecke algebra. In this talk, we show how these two notions can be generalized at the level of multipartitions in the context of Ariki-Koike algebras (joint work with M.Chlouveraki and C.Lecouvey)
  
  
• Frank Lübeck [RWTH Aachen]
  Titel: Green functions and permutation characters
  Abstract: Green functions of finite reductive groups are class functions on unipotent elements. They are used to compute the values of (ordinary) irreducible characters of these groups.
  
  Lusztig and Shoji decribed an algorithm for computing the Green functions as linear combinations of certain functions which are supported on elements in a single conjugacy class of the underlying algebraic group. At first these functions are only determined up to a root of unity factor, but these factors are now known in almost all cases.
  
  The missing cases are for groups of type \(E_8(q)\) in small characteristic. In this talk we sketch a solution for the missing cases by computing values of permutation characters of parabolic subgroups. This extends an idea of Meinolf Geck.
  
  
• Alexander Miller [Universität Wien]
  Titel: Foulkes characters
  Abstract: I will talk about certain characters of \(S_n\) called Foulkes characters. I will describe a geometric approach that works for several other reflection groups and elucidates all of the classical properties. I will also discuss several new results for these characters and their generalizations, along with some recent connections, including the connection with adding random numbers.
  
  
• Benjamin Sambale [Leibniz Universität Hannover]
  Titel: Sylow structure from the character table
  Abstract: By the (finally proven) height zero conjecture the character table of a finite group \(G\) determines whether \(G\) has an abelian Sylow \(p\)-subgroup \(P\). We show that this remains true if \(P\) is "close to" or "far from" abelian. More precisely, each of the following properties is detected from the character table: \(|P:P'|= p^2\), \(|P:Z(P)|=p^2\), \(P\) has maximal nilpotency class, \(P\) is minimal non-abelian. This is joint work with G. Navarro and A. Moretó.
  
  
• Damiano Rossi [City, University of London]
  Titel: Alternating sums for unipotent characters
  Abstract: Unipotent characters play a fundamental role in representation theory of finite reductive groups. Recently, Broué has suggested a vanishing statement for alternating sums involving unipotent characters in the spirit of Dade's conjecture. We answer Broué's question in the affirmative for linear and symplectic groups.
  
  
• Lucas Ruhstorfer [Bergische Universität Wuppertal]
  Titel: The Alperin--McKay conjecture and groups of Lie type \(B\) and \(C\)
  Abstract: The Alperin--McKay conjecture is a local-global conjecture in the modular representation theory of finite groups. To prove this conjecture it suffices to verify the so-called inductive Alperin--McKay condition for all finite simple groups. In this talk, we will discuss this inductive condition for simple groups of Lie type $B$ and \(C\). This is joint work with Julian Brough.