TALKS / ABSTRACTS

• Robert Boltje [UC Santa Cruz]
  Titel: The trivial source ring, coherent characters, and orthogonal units
  Abstract: Trivial source modules of a finite group \(G\) over a field \(F\) of characteristic \(p>0\) are direct summands of finite-dimensional permutation \(FG\)-modules. Their isomorphism classes form a monoid under the direct sum and tensor product operations. The associated Grothendieck ring is called the trivial source ring, \(T(FG)\). We give a description of the trivial source ring as a subring of a product of character rings. This allows us to characterize the group of orthogonal units (or equivalently, torsion units) of \(T(FG)\) as a direct product of two subgroups. The first factor is the unit group of the Burnside ring of the \(p\)-fusion system of \(G\), and the second factor consists of "coherent" linear characters on normalizers of \(p\)-subgroups. The main motivation of studying the torsion unit group of \(T(FG)\) is its connection with the group of \(p\)-permutation self-equivalences of \(FG\). This is joint work with Robert Carman.
  
  
• Thomas Breuer [RWTH Aachen]
  Titel: Compute reductions of character values to finite fields, or avoid this.
  Abstract: Several representation theoretic questions involve the reduction of (Brauer) character values to finite fields. I will talk about technical subtleties of some of these questions.
  
  
• Olivier Dudas [Université de Paris]
  Titel: Braid group actions for Deligne--Lusztig varieties
  Abstract: (Joint work in progress with C. Bonnafé, M. Broué, G. Malle, J. Michel, R. Rouquier)
  The irreducible characters of a finite reductive group \(G(q)\) (such as \(GL(n,q)\) or \(S0(n,q)\) are constructed using algebraic varieties attached to the elements of the Weyl group of \(G\), or even to the elements of its braid group. In this talk I will explain how one can also construct interesting braid group actions on these varieties or at least on their cohomology, extending work of Broué-Michel. I will show some rather strange properties of this action, and explain how it can be used to recover Lusztig's exotic Fourier transform.
  
  
• Jonas Hetz [Universität Stuttgart]
  Title: Unipotent character sheaves and character values of finite groups of Lie type
  Abstract: Let \(G\) be a finite group of Lie type. In order to tackle the problem of determining the character table of \(G\), Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases of the space of class functions on \(G\), one of them consisting of the irreducible characters, the other one of characteristic functions of Frobenius-invariant character sheaves. We will indicate how this problem can in principle be reduced to considering cuspidal character sheaves of (quasi-)simple groups and report on some recent progress in this area.
  
  
• Birte Johansson [TU Kaiserslautern]
  Titel: On the inductive condition for the McKay--Navarro conjecture for the Suzuki and Ree groups
  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 proved a reduction theorem and gave an inductive condition for it. In this talk, I give the ideas for the verification of the equivariance part of the inductive condition for the Suzuki and Ree groups in non-defining characteristic.
  
  
• Caroline Lassueur [TU Kaiserslautern]
  Title: On the trivial source character tables of \(\text{SL}_2(q)\)
  Abstract: Trivial source modules play an important rôle in block theory, for example in the description of various equivalences between block algebras. A remarkable property of these modules lies in the fact that they lift from positive characteritic to characteristic zero and hence can be studied through ordinary characters. The trivial source character table (also called species table of the trivial source ring) contains a lot of information about these modules and their associoated Brauer quotients. I will discuss recent results about these for the infinite family of finite groups \(\text{SL}_2(q)\) over a large enough field \(k\) of positive characteristic \(\ell\) via block-theoretical and character-theoretical methods. This is joint work with Niamh Farrell and Bernhard Böhmler.
  
  
• Yanjun Liu [Jiangxi Normal University/TU Kaiserslautern]
  Title: Higher Frobenius-Schur indicators
  
  Abstract: Similarly to the Frobenius-Schur indicator of irreducible characters, we consider higher Frobenius-Schur indicators of generalized characters of a finite group with respect to a prime power. It turns out that these invariants give answers to interesting questions in representation theory. In particular, we give several characterizations of groups via higher Frobenius-Schur indicators. This is a joint work with W. Willems.
  
  
• Frank Lübeck [RWTH Aachen]
  Title: Standardization of finite fields and their cyclic subgroups
  Abstract: Many computer algebra systems use Conway polynomials to define not too big finite fields. In such cases it is easy to exchange data about finite fields between systems. Furthermore, Conway polynomials are used for the computation of explicit Brauer characters of representations of finite groups in prime characteristic. Unfortunately, it is practically impossible to compute any Conway polynomials which are not already known while for applications it would be desirable to know them.
  In this talk I will sketch a proposal for a different standardization of finite fields with some nice properties. Furthermore standardized generators of cyclic (multiplicative) subgroups can be defined. Using these proposed definitions both applications mentioned above could be extended to a significantly wider range of finite fields in practice.
  
  
• Laura Voggesberger [TU Kaiserslautern]
  Title: Nilpotent pieces of Lie algebras of type \(\text{G}_2\) and \(\text{F}_4\) in bad characteristic
  Abstract: Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\), and let \(mfg\) be its Lie algebra. The partition of the unipotent variety of \(G\) defined by George Lusztig in his papers Unipotent elements in small characteristic I-IV is very useful when working with representations of \(G\). Alternatively, one can consider certain subsets of the nilpotent variety of \(mfg\) called pieces. This approach also appears in Unipotent elements in small characteristic III where the pieces in good characteristic (for all types) and for classical groups (in every characteristic) have already been computed.
  However, it is not yet known what the pieces for the exceptional groups \(G_2,F_4, E_6, E_7, E_8\) in bad characteristic are and whether they form a partition of the nilpotent variety as is the case in good characteristic.
  This talk will give an introduction to the theory of the nilpotent pieces and give some context for the unknown cases. Moreover, the solution for the pieces in type \(G_2\) and \(F_4\) in bad characteristic will be presented.