UC Santa Cruz

Let F be an algebraically closed field of positive characteristic p. Let G and H be finite groups. Let A be a block of FG and let B be a block of FH. A p-permutation equivalence between A and B is an element \gamma in the group of (A, B)-p-permutation bimodules with twisted diagonal vertices such that \gamma\cdot_H\gamma^o = [A] and \gamma^o\cdot_G\gamma = [B]. A p-permutation equivalence lies between a splendid Rickard equivalence and an isotypy.

We introduce the notion of a \gamma-Brauer pair, which generalizes the notion of a Brauer pair for a p-block of a finite group. The \gamma-Brauer pairs satisfy an appropriate Sylow theorem. Furthermore, each maximal \gamma-Brauer pair identities the defect groups, fusion systems and Külshammer-Puig classes of A and B. Additionally, the Brauer construction applied to \gamma induces a p-permutation equivalence at the local level, and a splendid Morita equivalence between the Brauer correspondents of A and B.