Graduate Colloquium Winter 2014

In this talk we describe joint work with Robert Boltje. 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 y in the group of (A, B)-p-permutation bimodules with twisted diagonal vertices such that y H Ÿ y ° = [A] and y °G y = [B]. A p-permutation equivalence lies between a splendid Rickard equivalence and an isotypy.

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