On Perfect Isometries for Tame Blocks

Any 2-block of a finite group G with a quaternion defect groupQ₈ is Morita equivalent to the corresponding block of the centraliserH of the unique involution of Q₈ in G; this answers positivelyan earlier question raised by M. Broué. 2000 MathematicsSubject Classification 20C20.     

Quillen’s stratification for fusion systems

The purpose of this note is to provide a reference for the fact that the proof of Quillen’s stratification for finite group cohomology carries over to fusion system. As in the case of Quillen’s stratification for block varieties, the proof is similar to the usual proof for group cohomology except for the use of fusion stable bisets, whose existence is due to Broto et al.     

Alperin’s weight conjecture in terms of equivariant Bredon cohomology

Alperins weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in §1 and §2, respectively, and §3 provides a tool for computing the cohomology of the functors arising in §2. Taking as starting point the alternating sum formulation of Alperins weight conjecture by Knörr-Robinson [11], the material of the previous sections is applied in §4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17].     

The strong Frobenius numbers for cyclic defect blocks are equal to one

Abstract

The purpose of this note is to provide a reference for the fact that the strong Frobenius number, in the sense of Eaton and Livesey, of a block of a finite group with a cyclic defect group is equal to 1. This answers a question of Farrell and Kessar.     

Blocks of minimal dimension

Any block with defect group P of a finite group G with Sylow-p-subgroup S has dimension at least |S|²/|P|; we show that a block which attains this bound is nilpotent, answering a question of G. R. Robinson. Received: 20 November 2006