Schedule
Tuesday, November 19, 2024
| 16:15 – 17:15 | Jonathan Gruber (prix Schläfli 2024) |
| followed by a prize ceremony and apéritif |
Wednesday, November 20, 2024
| 09:00 – 09:45 | Thomas Gerber |
| 10:00 – 10:30 | Gaëtan Mancini |
| 10:30 – 11:15 | coffee break |
| 11:15 – 12:00 | Filippo Ambrosio |
| 12:15 – 12:45 | Adam Thomas |
| 14:30 – 15:15 | coffee |
| 15:15 – 16:00 | Marie Roth |
| 16:15 – 17:00 | Martin Liebeck |
| 19:00 | conference dinner |
Thursday, November 21, 2024
| 09:00 – 09:45 | Mikko Korhonen |
| 10:00 – 10:30 | Iulian Simion |
| 10:30 – 11:00 | coffee break |
| 11:00 – 11:45 | Inna Capdeboscq |
| 12:00 – 12:30 | Gunter Malle |
Abstracts
Filippo Ambrosio
Étale geometry of Jordan classes in prime characteristic
Let G be a connected reductive algebraic group over an algebraically closed field k. Lusztig (1984) defined Jordan classes of G: they form a partition of the G into finitely many smooth subvarieties consisting of conjugacy classes of constant dimension. Particular cases of such varieties are the unipotent classes and the subset of regular semisimple elements; in general, Jordan classes have important applications to representation theory (description of sheets with respect to the conjugacy action, description of the generalized Springer correspondence). An analogue partition of the Lie algebra Lie(G) into subvarieties called decomposition classes (Zerlegungsklassen) dates back to Borho–Kraft (1979). These share several properties with their counterparts in G.
When k = C it is possible to reduce the study of geometric properties (e.g., smoothness) of a point g in the closure of a Jordan class J ⊂ G to the study of the geometry of an element x in the closure of the union of finitely many decomposition classes in Lie(M), where M ≤ G is a connected reductive subgroup depending on g. This talks aims at generalizing this reduction procedure to the case char k > 0 and at explaining conditions on chark for our results to hold.
Inna Capdeboscq
Groups of mixed characteristic in the Generation-2 proof of the Classification of Finite Simple Groups.
Let p be an odd prime. In this talk we will discuss classification of groups of simultaneously even- and p-types as it appears in the GLS-project. This is a joint work with R. Lyons and R. Solomon.
Thomas Gerber
Generalised Mullineux involution
In modular representation theory, the Mullineux involution is a combinatorial map describing the effect of tensoring an irreducible representation of the symmetric group with the sign representation.
Since its original conjectural description in 1979, constant progress has been made towards computing, interpreting, and generalising the Mullineux map, with a variety of successful applications.
In this talk, I will recall some of these aspects, and I will mention a new conjecture for computing the Mullineux involution.
Jonathan Gruber
Generic direct summands of tensor products
The representation theory of groups over fields of positive characteristic is often studied by decomposing the category of representations as a product of subcategories called “blocks”. In many cases, one can obtain stronger and more fine-grained results by considering one block of the category at a time. Unfortunately, this strategy is a priori not well suited for understanding the structure of tensor products of representations, because the tensor product of two representations in a given block may have direct summands in many different blocks. In this talk, I will explain how this problem can be (partially) resolved for categories of representations of simple algebraic groups. I will also discuss the existence of a new class of indecomposable representations, called “generic direct summands”, which appear generically as direct summands of tensor products of simple representations in different blocks.
Mikko Korhonen
Maximal solvable subgroups
A subgroup of a group G is said to be maximal solvable if it is maximal among the solvable subgroups of G. In his 1870 Traité, Jordan gave a classification of the maximal solvable subgroups of symmetric groups.
The classification reduces to the primitive case, which is equivalent to the problem of classifying maximal irreducible solvable subgroups of GL(d,p), where p is a prime. In GL(d,p), the problem is reduced to the case of primitive irreducible solvable subgroups. These subgroups are then constructed in terms of maximal irreducible solvable subgroups of general symplectic groups GSp(2k,r) (r prime) and orthogonal groups O^±(2k,2).
In this talk, we will discuss Jordan’s classification in modern terms. More generally, we consider the complete classification of maximal irreducible solvable subgroups of classical groups such as GL(n,q), GSp(n,q), and GO(n,q), where q is a power of a prime. We will also discuss the analogous problem for linear algebraic groups over algebraically closed fields.
Martin Liebeck
An application of representation theory to regular semigroups
In the theory of semigroups, one of the main objects of study are (von Neumann) regular semigroups, and a natural problem arising in the theory of regular matrix semigroups is the following, posed by Araujo and Cameron. Characterize the subgroups G of GL(n,q) with the property that for all singular matrices A, there exists g in G such that rank(A) = rank(AgA). I will show how this leads to a natural problem in group representation theory, the solution of which gives rise to some rather nice families of examples of such groups G.
Gunter Malle
Picky unipotent elements
In this talk we consider a question raised by Maróti, Martínez and Moretó on redundant Sylow subgroups in finite groups and the related notion of picky elements. We classify unipotent picky elements in reductive algebraic groups and in finite groups of Lie type.
Gaëtan Mancini
Multiplicity free tensor products for SL_3 and Sp_4
Let G be a simple algebraic group. A G-module is called multiplicity-free if all its composition factors are distinct. In this talk, we will discuss techniques for classifying multiplicity-free tensor products of simple modules, and we will show how this notion can give us information about the structure of these tensor products. As an application of these methods, we will discuss the classification in the case of SL_3 and Sp_4.
Marie Roth
On the unitriangularity of decomposition matrices of finite reductive groups
In 2020, Brunat-Dudas-Taylor showed that the decomposition matrix of the unipotent l-blocks of a finite reductive group G in good characteristic has unitriangular shape. Their theorem holds under some conditions on the prime l, in particular l being good. In this talk, we will discuss how to extend this result, firstly to l bad (for any G simple) and then to other blocks, namely the isolated blocks (for G simple of type G_2 and F_4). This work is part of my PhD thesis under the supervision of Gunter Malle and Olivier Dudas.
Iulian Simion
Variations of Landau’s theorem on conjugacy classes
In 1903 Landau showed that for a given k there are only finitely many groups having k conjugacy classes. Consequently, this led to a search for explicit lower bounds on the number of conjugacy classes in terms of the order of the group. More recently, Héthelyi and Külshammer showed that for a given k there are only finitely many groups having k conjugacy classes of prime power order. After mentioning some milestone results in this context, we comment on recent developments with focus on simple groups of Lie type. This is part of joint work with B. Çinarci, T. Keller and A. Maróti.
Adam Thomas
Epimorphic subgroups of simple algebraic groups
A subgroup H of an algebraic group G is epimorphic if every morphism phi: G –> G’ is uniquely determined by its restriction to H. There are many equivalent definitions and after discussing them I will briefly point to connections with algebraic geometry. Of particular relevance to us is a construction of epimorphic subgroups given by Bien and Borel. I will discuss ongoing joint work with Donna Testerman where we look to find the ingredients and follow this recipe for the simple algebraic groups in positive characteristic.