Inverse images of block varieties

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.

Weights for symmetric and general linear groups

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

Fusion category algebras

Fusion category algebras

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

Notes on p-blocks of characters of finite groups

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

Let B be a p-block of a finite group G with abelian defect group D such that S ≤ G ≤ Aut (S), S′ = S and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N G(D) in the sense of Broué. This has been done by Rouquier for principal blocks and it remains to deal with the non-principal blocks.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

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.

A note on Olssonʼs Conjecture

Cartan invariants and central ideals of group algebras

It is shown that ordinary irreducible characters of height zero in 2-blocks of finite groups have trivial 2-local Schur index. In view of Brauer's height zero conjecture, it is therefore extremely plausible that the irreducible characters in 2-blocks with abelian defect groups have trivial 2-local Schur index. If q is an odd prime, then there is no relationship expected between the height of an irreducible character in a q-block and the q-local Schur index, but a condition for a prime p≠q not to divide this index is given in terms of a rationality property of a defect class of the character's p-block.