Strong fusion control and stable equivalences

Abstract

This article is dedicated to the proof of the following theorem. Let G be a finite group, p be a prime number, and e be a p-block of G. Assume that the centraliser CG(P) of an e-subpair (P,eP) “strongly” controls the fusion of the block e, and that a defect group of e is either abelian or (for odd p) has a non-cyclic centre. Then there exists a stable equivalence of Morita type between the block algebras OGe and OCG(P)eP, where O is a complete discrete valuation ring of residual characteristic p. This stable equivalence is constructed by gluing together a family of local Morita equivalences, which are induced by bimodules with fusion-stable endo-permutation sources.Broué had previously obtained a similar result for principal blocks, in relation with the search for a modular proof of the odd Zp⁎-theorem. Thus our theorem points towards a block-theoretic analogue of the Zp⁎-theorem, which we state in terms of fusion control and Morita equivalences.