TALKS / ABSTRACTS

• Marc Cabanes [Université Paris Cité]
  Titel: On the McKay Conjecture
  Abstract: John McKay’s Conjecture (from 1971) predicts that, for any finite group G and prime \(\ell\), the number of complex irreducible representations of G with a degree not divisible by \(\ell\) is controlled by the normaliser of a Sylow \(\ell\)-subgroup of \(G\). By work of Isaacs, Malle and Navarro this conjecture was reduced in 2007 to a statement on finite quasisimple groups and their representation theory. After a series of results by various authors it is sufficient to verify a statement on quasisimple groups of type \(D\), namely Spin groups is left open. In the talk I will report on recent progress (joint work with B. Späth).
  
  
• Eirini Chavli [Universität Stuttgart]
  Title: The center of walled Brauer algebras
  Abstract: In 2015 Jung und Kim introduce a family of commuting elements of the walled Brauer algebra \(B_{r,s}(\delta)\), called the Jucys-Murphy elements. As similar in the case of symmetric groups, the supersymmetric polynomials in the Jucys-Murphy elements belong to the center of the walled Brauer algebra. They proved that, if the walled Brauer algebra is semisimple, then these supersymmetric polynomials generate the center and they conjectured the same for the non semisimple case. This is talk, we explain a general approach of calculating the center, which covers the semisimple and the non semisimple case. This approach allow us to prove the aforementioned conjecture for the case of \(B_{r,1}(\delta)\). This joint work with Maud de Visser, Alison Parker, Sarah Salmon, Urlica Wilson.
  
  
• Segio David Cia Luvecce [Sorbonne Université]
  Title: On a formula for the Lusztig induction over the unipotent representations of \(\operatorname{O}_{2n}^{\pm}(\mathbb F_q)\)
  Abstract: Asai gave an explicit formula for the Deligne-Lusztig induction of unipotent characters of the even special orthogonal groups. In particular, the induction to \(G^{\circ}=\operatorname{SO}_{2n+2e}^{\eta}(\mathbb F_q)\) (with \(\eta\in\{+,-\}\)) from a Levi subgroup isomorphic to \(T\times \operatorname{SO}_{2n}^{-\eta}(\mathbb F_q)\) (with \( |T|=q^e+1\)) of a unipotent irreducible character can be described in terms of Lusztig symbols, in a purely combinatorial way. In this talk, we will see the progress made on obtaining an analogue formula for the even orthogonal groups.
  
  
• Jonas Hetz [RWTH Aachen]
  Titel: The values of unipotent characters at unipotent elements for groups of type \(E_8\)
  Abstract: In order to tackle the problem of generically determining the character tables of the finite groups of Lie type \(\mathbf{G}(q)\) associated to a connected reductive group \(\mathbf{G}\) over \(\overline{F}_p\), Lusztig developed the theory of character sheaves in the 1980s. The subsequent work of Lusztig and Shoji in principle reduces this problem to specifying certain roots of unity. The situation is particularly well understood as far as character values at unipotent elements are concerned. We show how the explicit values of the unipotent characters at unipotent elements for the groups \(E_8(q)\) are obtained, by determining the respective roots of unity mentioned above.
  
  
• Linda Hoyer [RWTH Aachen]
  Titel: Oddness of orthogonal determinants of even-dimensional Specht modules
  Abstract: Let \(G\) be a finite group and \(\rho:G \to \mathrm{GL}_n(\mathbb{Q})\) be an absolutely irreducible representation. Let \(B\) be a \(\rho(G)\)-invariant, non-degenerate, symmetric bilinear form on \(\mathbb{Q}^n\). If \(n\) is even, the rational square class of the determinant of the Gram matrix of \(B\) can be uniquely represented by a positive integer \(d_{\rho}\). Richard Parker conjectures that this integer is always odd.
  In this talk, we will give a sketch of the proof of this conjecture for symmetric groups.
  
  
• Noelia Rizo [University of Valencia]
  Titel: Minimal heights and defect groups with two character degrees
  Abstract: One of the main problems of the past decades in representation theory of finite groups is Brauer's Height Zero Conjecture, posed by Richard Brauer in 1955. Now that the proof of this conjecture has recently been completed (Malle--Navarro--Schaeffer-Fry--Tiep, 2022), it seems appropriate to study possible extensions to blocks with non-abelian defect groups.
  
  In this sense, a conjecture proposed by C. Eaton and A. Moretó predicts that the smallest positive height of the irreducible characters in a \(p\)-block of a finite group and the smallest positive height of the irreducible characters in its defect group are equal. Very little is known on this problem. One inequality was shown to be a consequence of Dade's Projective Conjecture (Eaton--Moretó, 2014). In this occasion we will talk about a joint work with G. Malle and A. Moretó [1], where we prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
  
  References
  [1] G. Malle, A. Moretó, N. Rizo, Minimal heights and defect groups with two character degrees, arXiv:2305.19816.
  
  
• Marie Roth [RPTU Kaiserslautern-Landau]
  Title: Character sheaves of the principal series restricted to a conjugacy class
  Abstract: Let \({\mathbf G}\) be a connected reductive algebraic group. In this talk, we will consider the restriction of a character sheaf of \({\mathbf G}\) coming from a maximal torus to a conjugacy class of \({\mathbf G}\). Using this, one could extend the result of Brunat-Dudas-Taylor (2020) on the unitriangularity of decomposition matrices of the unipotent \(\ell\)-blocks to adjoint simple exceptional groups of Lie type, when \(\ell\) is bad.
  
  
• Jay Taylor [University of Manchester]
  Titel: Real Elements in Finite Reductive Groups
  Abstract: An element of a finite group is said to be real if it is conjugate to its inverse. In this talk we will discuss the problem of determining the real elements in a finite reductive group. We'll consider groups of type E6 as a running example.
  
  
• Laura Voggesberger [Ruhr-Universität Bochum]
  Title: Maximal Subgroups of Connected Reductive Groups
  Abstract: Let \(k\) be any field and let \(G\) be a connected reductive algebraic \(k\)-group. Associated to \(G\) is an invariant called the index of \(G\) (a Dynkin diagram along with some additional combinatorial information). Tits showed that the \(k\)-isogeny class of \(G\) is uniquely determined by its index and the \(k\)-isogeny class of its anisotropic kernel.
  Let \(H\) be a connected reductive \(k\)-subgroup of maximal rank in \(G\). One can define an invariant of the \(G(k)\)-conjugacy class of \(H\) in \(G\) called the embedding of indices of \(H\subset G\). Using this, one can begin to classify the maximal connected subgroups of maximal rank in \(G\) up to an invariant called "index-conjugacy" for any arbitrary field \(k\) and \(G\) absolutely simple. This has been done for absolutely simple groups of exceptional type by D. Sercombe. We will continue this work for absolutely simple groups of classical type. This is joint work with Damian Sercombe.