Program

Opening

Frank Lübeck

Title:  TBA

Abstract: TBA

Eirini Chavli

Title:  Interval Garside groups arising from involutions

Abstract:  Interval Garside groups arise from suitable intervals in the absolute order on finite reflection groups. While intervals associated with Coxeter elements lead to the well-known dual Artin groups, we investigate the groups obtained from intervals associated with involutions. We describe the resulting Garside groups and show that, except in type (Bn), they are isomorphic to (possibly decomposable) right-angled Artin groups. We also discuss some examples arising from finite complex reflection groups.

This is joint work with Thomas Gobet.

Mátyás Domokos

Title:  Degree bounds for polynomial invariants of finite groups

Abstract: A classical theorem of Emmy Noether asserts that the algebra of invariant polynomial functions on a finite dimensional complex vector space endowed with a linear action of a finite group is generated by homogeneous elements whose degree is bounded by the order of the group. Motivated by this result, the maximal possible degree of an indecomposable invariant of a given finite group is called the Noether number of the group. In the talk we shall survey some results relating the Noether number and its variants that received attention in recent years.

Gerhard Röhrle

Title:  G-complete reducibility and semisimplification

Abstract:  Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If G = GLn , then there is a degeneration process for obtaining from H a completely reducible subgroup H′ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility due to Serre and results from geometric invariant theory over non-algebraically closed fields. Our construction produces a G-completely reducible subgroup H′ of G, unique up to G(k)-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for G = GLn and with Serre’s ‘G-analogue’ of semisimplification for subgroups of G(k)). Time permitting, I shall show that under some extra hypotheses, one can pick H′ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau. This is a report on earlier joint work with Michael Bate and Benjamin Martin.

Carolina Vallejo

Title:  Fields of Values of Characters and Degree Constraints

Abstract:  In this talk, we discuss the abelian number fields arising as fields of values of irreducible characters of finite groups under degree constraints relative to a fixed prime p. 

This is based on joint work with G. Navarro, L. Ruhstorfer and P. H. Tiep.

Chaithra Pilakkat

Title:  Marked multi-colorings and root multiplicities of BKM superalgebras

Abstract:  In this talk, we present explicit combinatorial formulas for the root multiplicities of Borcherds–Kac–Moody (BKM) superalgebras, derived from their denominator identities and expressed in terms of graph invariants associated with marked (quasi) Dynkin diagrams. A central notion in our approach is that of marked multi-colorings and their associated polynomials, which generalize chromatic polynomials and provide an effective framework for computing root multiplicities.

We also introduce partially commutative Lie superalgebras (PCLSAs) as a tool for studying certain roots of BKM Lie superalgebras. We present a direct combinatorial proof of their denominator identity using ideas from Viennot’s heap theory. We then characterize the roots of PCLSAs and establish connections between their universal enveloping algebras and right-angled Coxeter groups, which lead to explicit formulas for their Hilbert series. This talk is based on joint work with Deniz Kus and R. Venkatesh.

Caroline Lassueur

Title:  Classifying tame blocks of maximal defect up to splendid Morita equivalence

Abstract:  In this talk, I will report on recent results obtained with S. Koshitani,
İ. Tuvay and D. Yılmaz on the classification of maximal-defect 2-blocks of tame representation type up to splendid Morita equivalence.

Beyond a complete classification result in the dihedral case, the most interesting feature of this work is certainly the variety of methods required in its proof. I will focus on describing how our arguments combine classical techniques of the block theory with a broad range of tools and results that have emerged over the past twenty-five years.  Amongst others, these include in a crucial manner Gorenstein-Walter’s classification of finite groups with dihedral Sylow 2-subgroups, the Erdmann-McGregor classification of tame blocks of finite groups, Linckelmann’s glueing of local Morita equivalences to obtain a stable equivalence of Morita type, the modern versions of Fong’s 1st and 2nd reductions in terms of splendid and endo-permutation source Morita equivalences, the structure of the Dade group of endo-permutation modules, Boltje-Perepelitsky’s extended theory of Brauer pairs for p-permutation modules, the structure and characters of trivial source modules, or the fusion systems of the blocks involved. Finally, the GAP Character Table Library allows us to pull characteristic 2 problems back to characteristic zero character-theoretical computations.

Gerhard Hiss

Title:  A global bound for decomposition numbers of unipotent blocks

Abstract:  This is a report on joint work with Radha Kessar. Arnaud Eteve recently proved a conjecture of Dudas and Malle on intersection cohomology characters of finite groups of Lie type. This yields upper bounds on the decomposition numbers of unipotent blocks of simple groups of Lie type.