Research

My main research interests lie in representation and character theory of finite groups. In particular, I am interested in the so-called Local-Global conjectures in modular representation theory of finite groups with a focus on the verification of the inductive conditions for quasi-simple groups of Lie type (a.k.a. finite reductive groups). This is an accessible introduction to group theory and representation theory written by Eugenio Giannelli and Jay Taylor. A more specialized survey on Local-Global conjectures written by Gunter Malle can be found here. This note,  written by Radha Kessar and Gunter Malle, describes some recent achievements in this research area by focusing on the crucial case of simple groups.

Publications

Submitted

Abstract: In this paper, we show that the Character Triple Conjecture holds for all finite groups once assumed for all quasi-simple groups. This answers the question on the existence of a self-reducing form of Dade's conjecture, a problem that was long investigated by Dade in the 1990s. Our result shows that this role is played by the Character Triple Conjecture, recently introduced by Späth, that we present here in a general form free of all previously imposed restrictions.

Abstract: Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that assuming the so-called Inductive (Blockwise) Alperin Weight Condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin Weight Conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro's conjecture for each finite group.

Abstract: We prove that Späth's Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin-McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer's Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at the prime 3.

Abstract: In this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case, the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalent to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard. 

Abstract: We study the Brown complex associated to the poset of l-subgroups in the case of a finite reductive group defined over a field F_q of characteristic prime to l. First, under suitable hypotheses, we show that its homotopy type is determined by the generic Sylow theory developed by Broué and Malle and, in particular, only depends on the multiplicative order of q modulo l. This result leads to several interesting applications to generic Sylow theory, mod l homology decompositions, and l-modular representation theory. Then, we conduct a more detailed study of the Brown complex in order to establish an explicit connection between the local-global conjectures in representation theory of finite groups and the generic Sylow theory. This is done by isolating a family of l-subgroups of finite reductive groups that correspond bijectively to the structures controlled by the generic Sylow theory.

Abstract: We obtain an adaptation of Dade's Conjecture and Späth's Character Triple Conjecture to unipotent characters of simple, simply connected finite reductive groups of type A, B and C. In particular, this gives a precise formula for counting the number of unipotent characters of each defect d in any Brauer l-block B in terms of local invariants associated to e-local structures. This provides a geometric version of the local-global principle in representation theory of finite groups. A key ingredient in our proof is the construction of certain parametrisations of unipotent generalised Harish-Chandra series that are compatible with isomorphisms of character triples.

Abstract: We work towards a version of generalized Harish-Chandra theory compatible with Clifford theory and with the action of automorphisms on irreducible characters. This provides a fundamental tool to verify the inductive conditions for the so-called Local-Global conjectures in representation theory of nite groups in the crucial case of groups of Lie type in non-defining characteristic. In particular, as shown by the author in an earlier paper, this has a strong impact on the verification of the inductive condition for Dade’s Conjecture. As a by-product, we also show how to extend the parametrization of generalised Harish-Chandra series given by Broué-Malle-Michel to the non-unipotent case by assuming maximal extendibility. 


Published

Abstract: We prove new results in generalised Harish-Chandrat theory providing a description of the so-called Brauer-Lusztig blocks in terms of information encoded in the l-adic cohomology of Deligne-Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the l-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between l-structures and their geometric counterpart. Finally, using the description of Brauer-Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisation of generalised Harish-Chandra series.

Abstract: Let p be a prime, B a p-block of a finite group G and b its Brauer correspondent. According to the Alperin–McKay Conjecture, there exists a bijection between the set of irreducible ordinary characters of height zero of B and those of b. In this paper, we show that whenever G is p-solvable such a bijection can be found, both for ordinary and Brauer characters, with the additional property of being compatible with divisibility of character degrees. In this case, we also show that the dimension of b divides the dimension of B.

Abstract: We give new evidence of the fact that the structure of a solvable group can be controlled by irreducible monomial characters. In particular, we inspect the role of monomial characters in the Isaacs-Navarro-Wolf Conjecture and in Gluck’s Conjecture.

Abstract: We refine the reduction theorem of the McKay Conjecture proved by Isaacs, Malle, and Navarro. Assuming the inductive McKay condition, we obtain a strong version of the McKay Conjecture that gives central isomorphic character triples.

Abstract: In this paper, we prove Späth's Character Triple Conjecture for p-solvable groups. This is a conjecture proposed by Späth during the reduction process of Dade's Projective Conjecture to quasisimple groups. In addition, as suggested by Isaacs and Navarro, we take into account the p-residue of characters.

Abstract: Let G be a p-solvable group, P a p-subgroup and χ in Irr(G) such that χ(1)_p ≥ |G:P|_p. We prove that the restriction χ_P is a sum of characters induced from subgroups Q≤  P such that χ(1)_p=|G:Q|_p. This generalizes previous results by Giannelli-Navarro and Giannelli-Sambale on the number of linear constituents of χ_P. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer--Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.

In preparation

Theses

PhD thesis

Title: Character Triple Conjecture: towards the inductive condition for Dade's Conjecture in nondefining characteristic

Defended: 2nd February 2022

Advisor: Britta Späth


Master's thesis

Title: Monomial characters of finite solvable groups

Defended: 18th July 2018

Advisor: Silvio Dolfi


Coauthors