The Broué invariant of a $p$-permutation equivalence

Abstract

A perfect isometry $I$ (introduced by Broué) between two blocks $B$ and $C$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $ψ$ of $C$ to $\pm$ an irreducible character of $B$. Broué proved that the ratio of the codegrees of $ψ$ and $I(ψ)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $ψ$. We call this element the Broué 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 Broué 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 Broué 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.