Abstracts

• Frank Himstedt [TU München]
  Title: On the characters of the Sylow \(p\)-subgroups of untwisted Chevalley groups \(G(p^a)\)
  Abstract:
  Let \(G(q)\) be an untwisted Chevalley group over a field with \(q\) elements and \(U(q)\) a Sylow \(p\)-subgroup of \(G(q)\) where \(p\) is the defining characteristic. The set of ordinary irreducible characters of \(U(q)\) can be partitioned into families such that each family is labeled by a certain set of positive roots (these roots describe the root subgroups that are contained in the center, but not in the kernel of the characters in the family). In my talk, I describe some results on those families that are labeled by a set consisting of only one root. This is joint work with Tung Le and Kay Magaard.


• Benedetta Lancellotti [Università degli Studi di Milano - Bicocca]
  Title: Linear source lattices and Alperin-McKay conjecture
  Abstract:
  In a recent preprint together with Shigeo Koshitani and Thomas Weigel, we establish a connection between the Alperin-McKay conjecture for a finite group \(G\) and a prime number \(p\) and the Grothendieck group \(\mathbf{L}_{\mathcal{O}}^{\mathrm{mx}}(B)\) of linear source \({\mathcal{O}} G\)-lattices of maximal vertex in a \(p\)-block \(B\), where \({\mathcal{O}}\) is a suitable complete discrete valuation ring of characteristic \(0\) with quotient field \(\mathbb{K}\) and residue field \(\mathbb{F}\) of characteristic \(p\). Moreover, if a particular (splitting) modular system is involved, also the Isaacs-Navarro refinement of Alperin-McKay conjecture appears strictly linked to linear source lattices.


• Michael Livesey [University of Manchester]
  Title: Towards Donovan's conjecture for abelian defect groups
  Abstract:
  We introduce a new invariant for a \(p\)-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index \(p\). As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for blocks with abelian \(2\)-groups of rank at most \(4\) as defect groups. We also apply these methods to completely classify all blocks with defect group \(C_{2^n}\times C_2\times C_2\) up to Morita equivalence. This is all joint work with Charles Eaton.


• Frank Lübeck [RWTH Aachen]
  Title: Parameterization of semisimple conjugacy classes
  Abstract:
  Let \(\{G(q)\mid q \textrm{ a prime power}\}\) be a series of groups of Lie type, were \(G\) has a fixed type and rank. To describe the conjugacy classes and ordinary irreducible representations (or their characters) of these groups we need as the first step a parameterization of their conjugacy classes (classes of elements with order prime to the defining characteristic).
  
  We assume that the series of groups is given by a simple combinatorial structure, called a root datum, together with an action on this datum by Frobenius morphisms. The basic strategy is to classify the possible centralizers of semisimple elements up to \(G(q)\)-conjugacy first (in terms of the root system and the Weyl group of \(G\)) and then to compute representatives of the classes with any of these centralizers.
  
  We aim for a generic parameterization which works for all groups in the considered series, that is with the field size \(q\) as a parameter.


• Kay Magaard [University of Birmingham]
  Title: Overgroups of Irreducible Quasisimple Subgroups in Finite Classical Groups
  Abstract:
  The primary open problem in the classification of the maximal subgroups of the finite almost simple groups is the determination of the overgroups of finite quasisimple groups which act irreducibly on the natural module of a finite classical group. We will discuss computatiional methods which can be brought to bear of this problem.


• Jean Michel [Université Denis Diderot - Paris 7]
  Title: Recent developments in Chevie
  Abstract:
  I will present Chevie by examples, emphasizing developments during the last year.


• Raphaël Rouquier [UCLA]
  Title: Computing perverse equivalences
  Abstract:
  Broué’s conjecture predicts the existence of a derived equivalence between a block with abelian defect groups and a Brauer correspondent. Perverse equivalences are a particular type of derived equivalences encoded by some combinatorial data. We will discuss the problem of computing perverse simple modules.
  For sporadic groups, the problem is to find a perversity function providing a lift of the given stable equivalence. For finite groups of Lie type, a different setting can be used, explaining genericity properties, and relying on invariant theory on polynomial and exterior algebras.
  This is joint work with David Craven.


• Lleonard Rubio y Degrassi [City, University of London]
  Title: On Hochschild cohomology and global/local structures
  Abstract:
  In this talk I will discuss the interplay between the local and the global invariants in modular representation theory with a focus on the first Hochschild cohomology \(\mathrm{HH}^1(B)\) of a block algebra \(B\).
  In particular, I will show the compatibility between \(r\)-integrable derivations and stable equivalences of Morita type. I will also show that if \(\mathrm{HH}^1(B)\) is a simple Lie algebra such that \(B\) has a unique isomorphism class of simple modules, then \(B\) is nilpotent with an elementary abelian defect group \(P\) of order at least \(3\). The second part is joint work with M. Linckelmann.


• Jay Taylor [University of Arizona]
  Title: Action of automorphisms on irreducible characters of symplectic groups
  Abstract:
  In recent years a new approach has been developed to several long standing conjectures in the representation theory of finite groups; such as the McKay conjecture. These conjectures are stated for all finite groups but the recent approach has reduced these conjectures to checking certain conditions on quasisimple finite groups (often referred to as inductive conditions). This, in theory, makes the conjecture more manageable thanks to the classification of finite simple groups. However, the downside to this is that one requires information about how automorphisms act on the irreducible characters of quasisimple finite groups. In this talk we present results in this direction concerning the finite symplectic groups Sp(2n,q) where q is an odd prime power. Specifically we completely describe the action of the automorphism group on the ordinary irreducible characters of these groups.


• Carolina Vallejo Rodríguez [ICMAT]
  Title: On Brauer correspondent blocks
  Abstract:
  Let G be a finite group and let F be an algebraically closed field of prime characteristic p. The group algebra FG decomposes as a sum of algebras, which are called the (p-)blocks of G. R. Brauer associated to each block B of G a block b of a local p-subgroup of G. One of the main problems in Group Representation Theory is to study the relationship between B and b (what properties these blocks share, etc.). In this talk, we will show how the Galois refinement of the Alperin-McKay conjecture, proposed by G. Navarro, can help us to partially understand this problem. (This is joint work with G. Navarro and P.H. Tiep).