Alperin Weight Conjecture Research Articles

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.

The Alperin Weight Conjecture and Dade's Conjecture for the Simple Group Fi′24

AbstractWe classify the radicalp-subgroups and chains of the Fischer simple group Fi′24and then verify the Alperin weight conjecture and the Uno reductive conjecture for Fi′24.

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.

The Alperin and Uno Conjectures for the Fischer Simple Group Fi22

ABSTRACT Using the local subgroup strategy of An and O'Brien (1997), An and O'Brien (1999), we classify the radical subgroups and chains of the Fischer simple group Fi 22 and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs–Navarro conjectures. #Communicated by B. Parshall.

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.

The Alperin and Dade Conjectures for the Conway Simple Group Co1

We classify the radical subgroups and chains of the Conway simple group Co1 and then verify the Alperin weight conjecture and the Dade final conjecture for this group.

The Alperin Weight Conjecture and Uno's Conjecture for the Baby Monster B, p Odd

AbstractSuppose that p is 3, 5 or 7. In this paper, faithful permutation representations of maximal p-local subgroups are constructed, and the radical p-chains of the Baby Monster B are classified. Hence, the Alperin weight conjecture and the Uno reductive conjecture can be verified for B, the latter being a refinement of Dade's reductive conjecture and the Isaacs-Navarro conjecture.

Conjectures on the character degrees of the harada-norton simple group HN

We classify the radical subgroups and chains of the Harada-Norton simple group HN and verify the Alperin weight conjecture and the refined Dade conjecture due to Uno for the group. This implies the Isaacs-Navarro and Dade reductive conjectures for the group.

The Alperin Weight Conjecture and Dade's Conjecture for the Simple Group J4

AbstractThe authors construct faithful permutation representations of maximal 2-local subgroups and classify the radical chains of the Janko simple group J4; hence the Alperin weight conjecture and the Dade reductive conjecture for J4 are verified.

The Alperin and Dade Conjectures for the O'Nan and Rudivalis Simple Groups

We classify the radical subgroups and chains of the O'Nan and Rudvalis simple groups O'N and Ru, and then verify the Alperin weight conjecture and the Dade final conjecture for the two groups.

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 the simple Conway’s third group

This paper is part of a program to study Alperin’s weight conjecture and Dade’s conjecture on counting ordinary characters in blocks for several finite groups. The classifications of radical subgroups and certain radical chains and their local structures of the simple Conway’s third group have been obtained by using the computer algebra system CAYLEY. The Alperin weight conjecture and the Dade final conjecture have been confirmed for the group.

THE ALPERIN AND DADE CONJECTURES FOR THE FISCHER SIMPLE GROUP Fi23

We modify the local strategy of [2] and use it to classify the radical subgroups and chains of the Fischer simple group Fi 23. We verify the Alperin weight conjecture and the Dade final conjecture for this group.

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.