The Brou\'e invariant of a $p$-permutation equivalence

A perfect isometry $I$ (introduced by Brou\'e) between two blocks $B$ and $C$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $\psi$ of $C$ to $\pm$ an irreducible character of $B$. Brou\'e proved that the ratio of the codegrees of $\psi$ and $I(\psi)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $\psi$. We call this element the Brou\'e invariant of $I$. The goal of this paper is to show that if $I$ comes from a $p$-permutation equivalence or a splendid Rickard equivalence between $B$ and $C$ then, up to a sign, the Brou\'e invariant of $I$ is determined by local data of $B$ and $C$ and therefore, up to a sign, is independent of the $p$-permutation equivalence or splendid Rickard equivalence. Apart from results on $p$-permutation equivalences, our proof requires new results on extended tensor products and bisets that are also proved in this paper. As application of the theorem on the Brou\'e invariant we show that various refinements of the Alperin-McKay Conjecture, introduced by Isaacs-Navarro, Navarro, and Turull are consequences of $p$-permutation equivalences or Rickard equivalences over a sufficiently large complete discrete valuation ring or over $\mathbb{Z}_p$, depending on the refinement.