| Résume | 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 $p$-\textit{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 $\gamma\cdot_H\gamma^{\circ}=[A]$ and $\gamma^{\circ}\cdot_G\gamma=[B]$. 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 identifies 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 $B.$ |
