Strong fusion control and stable equivalences

TL;DR

This paper proves a theorem establishing stable Morita equivalences between certain block algebras of finite groups under fusion control conditions, advancing the understanding of block theory and modular representation theory.

Contribution

It introduces a new stable equivalence construction for blocks with controlled fusion, generalizing Broué's results for principal blocks and suggesting a block-theoretic analogue of the Z*p-theorem.

Findings

Stable Morita equivalence between block algebras under fusion control.

Construction of equivalences via local Morita equivalences with fusion-stable sources.

Extension of Broué's results to broader classes of blocks.

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 C_G(P) of an e-subpair (P,e_P) "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 center. Then there exists a stable equivalence of Morita type between the block algebras OGe and OC_G(P)e_P, 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. Brou\'e had previously obtained a similar result for principal blocks, in relation with the search for a modular proof of the odd Z*p-theorem. Thus our theorem points towards a block-theoretic analogue of the Z*p-theorem, which we state in terms of fusion control and Morita equivalences.