Alperin Weight Research Articles

Modular representations of finite groups and Lie theory

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent.The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character.We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.

Weights for ℓ-local compact groups

In this note, we initiate the study of F-weights for an ℓ-local compact group F over a discrete ℓ-toral group S with discrete torus T. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when T is the unique maximal abelian subgroup of S up to F-conjugacy and every element of S is F-fused into T, the number of weights of F is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of F with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when F is simple and |S:T|=ℓ. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.

Alternating sums over π-subgroups

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p -defect is determined by local subgroups. In this paper we replace p by a set of primes π and prove a π -version of Dade's conjecture for π -separable groups. This extends the (known) p -solvable case of the original conjecture and relates to a π -version of Alperin's weight conjecture previously established by the authors.

Some results on a variant of the Green correspondence with applications to Alperin's weight conjecture for blocks

Some results on a variant of the Green correspondence with applications to Alperin's weight conjecture for blocks

Blocks with the hyperfocal subgroup [formula omitted

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G, whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has hyperfocal subgroup Z2n×Z2n for a positive integer number n, when the block is controlled by the normalizer NG(P) and the hyperfocal subgroup is contained in the center of P, or when the block is not controlled by NG(P) and the hyperfocal subgroup is contained in the center of the unique essential subgroup in the fusion system and has order at most 16. In particular, Alperin's weight conjecture holds in the considered cases.

On Alperin's weight conjecture for $p$-blocks of $p$-solvable groups

For $p$-solvable groups, a strong form of Alperin's weight conjecture has been proved by T. Okuyama (unpublished). L. Barker has refined this theorem by taking Green correspondence into account. We prove here a relative version of Barker's theorem.

Two remarks on the reduction of Alperin's weight conjecture

The so-called inductive McKay condition on finite simple groups, due to Isaacs–Malle–Navarro, has recently been reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro–Tiep. This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.

Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8

Using the classification of finite simple groups, we prove Alperin's weight conjecture and the character theoretic version of Broué's abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

On stable equivalences and blocks with one simple module

Using a stable equivalence due to Rouquier, we show that Alperin's weight conjecture holds for any p-block of a finite group with defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules.

Endomorphism rings of permutation modules

Let k be an algebraically closed field of characteristic p and G a finite group. Then the permutation module k P G of G on the cosets of a Sylow p -subgroup P is via Fitting correspondence strongly related to its endomorphism ring End k G ( k P G ) . On the other hand, each Green correspondent in G of a weight module of G occurs as a direct summand of k P G . This fact suggests to analyze both structures, the permutation module and the associated endomorphism ring towards hints at a proof for Alperin's weight conjecture. We present a selection of such investigations for different groups and characteristics. In particular we focus on the socle and head constituents of the indecomposable direct summands of k P G and of the PIMs of E E .

Standard stratifications of EI categories and Alperin's weight conjecture

We characterize the finite EI categories whose representations are standardly stratified with respect to the natural preorder on the simple representations. The orbit category of a finite group with respect to any set of subgroups is always such a category. Taking the subgroups to be the p-subgroups of the group, we reformulate Alperin's weight conjecture in terms of the standard and proper costandard representations of the orbit category. We do this using the properties of the Ringel dual construction and a theorem of Dlab, which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules. We indicate that these results hold in the generality of an algebra whose simple modules are preordered, rather than partially ordered.

Representations of Finite Groups

The meeting was organized by A. Kleshchev (Eugene), M. Linckelmann (Aberdeen), G. Malle (Kaiserslautern) and J. Rickard (Bristol). This meeting was attended by over 50 participants with broad geographic representation. It covered a wide variety of aspects of the representation theory of finite groups and related objects like Hecke algebras. This workshop was sponsored by a project of the European Union which allowed us to invite in addition to established researchers also a couple of young people working on a PhD in representation theory. In eleven longer lectures of 40 minutes each and twentytwo shorter contributions of 30 minutes each, recent progress in representation theory was presented and interesting new research directions were proposed. Besides the lectures, there was plenty of time for informal discussion between the participants, either continuing ongoing research cooperation or starting new projects. The topics of the talks came roughly from two major areas: on the one hand side, the investigation of representation theoretic properties of general finite groups and related objects, on the other hand the determination and detailed analysis of representations of special classes of finite groups and related objects like Hecke algebras. \par One of the main topics touched upon in several talks was the investigation of the various open conjectures on characters and blocks of finite groups, like Alperin's, Brou'e's and Dade's conjecture. A major breakthrough presented at this meeting by G. Navarro was the reduction (jointly with M. Isaacs and G. Malle) of the McKay conjecture on character degrees to a statement purely about simple groups, and the verification of this condition for certain families of finite simple groups. In the same direction, Puig announced some reduction statements for Alperin's weight conjecture.

Fusion category algebras

The fusion system F on a defect group P of a block b of a finite group G over a suitable p -adic ring O does not in general determine the number l ( b ) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k × on the orbit category F c of F -centric subgroups Q of P of b which “glues together” the second cohomology classes α ( Q ) of Aut F (Q) with values in k × in Külshammer–Puig [Invent. Math. 102 (1990) 17–71]. We show that if α exists, there is a canonical quasi-hereditary k -algebra F (b) such that Alperin's weight conjecture becomes equivalent to the equality l(b)=l( F (b)) . By work of C. Broto, R. Levi, B. Oliver [J. Amer. Math. Soc. 16 (2003) 779–856], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of F c by the center functor Z . If both invariants α , L exist, we show that there is an O -algebra L (b) associated with b having F (b) as quotient such that Alperin's weight conjecture becomes again equivalent to the equality l(b)=l( L (b)) ; furthermore, if b has an abelian defect group, L (b) is isomorphic to a source algebra of the Brauer correspondent of b .

Fusion category algebras

The fusion system F on a defect group P of a block b of a finite group G over a suitable p-adic ring O does not in general determine the number l( b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k × on the orbit category F c of F -centric subgroups Q of P of b which “glues together” the second cohomology classes α( Q) of Aut F (Q) with values in k × in Külshammer–Puig [Invent. Math. 102 (1990) 17–71]. We show that if α exists, there is a canonical quasi-hereditary k-algebra F (b) such that Alperin's weight conjecture becomes equivalent to the equality l(b)=l( F (b)) . By work of C. Broto, R. Levi, B. Oliver [J. Amer. Math. Soc. 16 (2003) 779–856], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of F c by the center functor Z . If both invariants α, L exist, we show that there is an O -algebra L(b) associated with b having F (b) as quotient such that Alperin's weight conjecture becomes again equivalent to the equality l(b)=l( L(b)) ; furthermore, if b has an abelian defect group, L(b) is isomorphic to a source algebra of the Brauer correspondent of b.

Some Simple Projective Brauer Quotients of Simple Modules for the Symmetric Groups in Characteristic Two

By Alperin's weight conjecture (J. L. Alperin and P. Fong, 1990, J. Algebra131, 2–22) the number of simple kSn-modules equals the number of weights for Sn, where Sn is the symmetric group on n symbols and k is a field of characteristic p>0. In this paper we answer the question, “When is the Brauer quotient of a simple F2Sn-module V with respect to a subgroup H of Sn both simple and projective as an NSn(H)/H-module?,” in some special cases. Remarkably, in each case there is only one such subgroup H (up to conjugacy).

The Alperin and Dade Conjectures for Ree Groups2F4(q2) in Non-defining Characteristics

This paper is part of a program to study Alperin's weight conjecture and Dade's ordinary conjecture on counting characters in blocks for several finite groups. The local structures of radical subgroups and certain radical 3-chains of a Ree group of typeF4are given and the conjectures and a conjecture of Olsson are confirmed for the group when the characteristic of the modular representation is distinct from the defining characteristic of the group.