Fusion category algebras

Fusion category algebras

For $p$-solvable groups, a strong form of Alperin's weight conjecture has been proved by T. Okuyama (unpublished).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.

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

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.

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. Kawai’s result is a tool to calculate the cohomology variety of a module in a block B of a finite group algebra kG restricted to subgroups of a defect group P, provided that P is abelian. Kawai’s result coincides with a Theorem of Avrunin and Scott specialized to modules in the principal block and their restrictions to p-subgroups. J. Rickard raised the question whether Kawai’s result can be extended to modules in blocks with arbitrary defect groups. We show that this is indeed the case for modules whose corresponding module over some almost source algebra is fusion stable. We show that this fusion stability hypothesis is automatically satisfied for principal blocks and blocks with abelian defect groups.

The 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.

Brouéʼs abelian defect group conjecture holds for the sporadic simple Conway group [formula omitted

In this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups.

The first author has recently classified the Morita equivalence classes of 2-blocks B of finite groups with elementary abelian defect group of order 32. In all but three cases he proved that the Morita equivalence class determines the inertial quotient of B. We finish the remaining cases by utilizing the theory of lower defect groups. As a corollary, we verify Broue's Abelian Defect Group Conjecture in this situation.

On p-blocks with abelian defect groups and inertial index 2 or 3, II

Perfect Isometries for Blocks with Abelian Defect Groups and Cyclic Inertial Quotients of Order 4

Notes on p-blocks of characters of finite groups

Abstract.We show that the blockwise version of the Alperin weight conjecture is true if for every finite non-abelian simple group a set of conditions holds. Furthermore we prove that several series of simple groups satisfy these assumptions. This refines recent work of Navarro and Tiep, who proved an analogous reduction theorem for the non-blockwise version of the Alperin weight conjecture.

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

ABSTRACTLet M be a semiprime Goldie module which is projective in σ[M]. We give several decompositions of the M-injective hull of M in terms of its minimal prime submodules. Then, we use this to give a decomposition of the endomorphism ring of . We also show that the set of minimal primes of M is in bijective correspondence with the set of isomorphism classes of simple modules over . We also investigate the relationships between semiprime Goldie modules, QI-modules and co-semisimple modules. As an application we extend certain results on left QI-rings and V-rings.