Broue's Abelian Defect Group Research Articles

Broué's conjecture for 2-blocks with elementary abelian defect groups of order 32

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.

Classifying blocks with abelian defect groups of rank 3 for the prime 2

In this paper we classify all blocks with defect group C2n×C2×C2 up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of 2-blocks with abelian defect groups of rank at most 3. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index 2, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above.

On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the Abelian defect group conjecture

Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras.

Broué’s abelian defect group conjecture for alternating groups

We establish Broue's abelian defect group conjecture for the alternating groups, using the Chuang-Rouquier theorem (proving this for the symmetric groups) and a descent result.