Fixed points of extended tensor products

TL;DR

This paper introduces new diagrams linking $p$-permutation equivalences to local stabilizer equivalences in finite group algebras, using advanced fixed point and tensor product formulas.

Contribution

It develops new technical tools for fixed points and tensor products that generalize previous results, enabling the analysis of $p$-permutation equivalences via Brauer constructions.

Findings

New formulas for fixed points and tensor products of group actions

Lifts of isotypy diagrams to local equivalences

Generalizations of Boltje-Danz and Boltje-Perepelitsky results

Abstract

For a $p$-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the $p$-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding Brauer pairs. These diagrams can be viewed as lifts of the square diagrams in the definition of isotypies. The proof of the commutativity requires new technical tools, namely a formula for how taking fixed points commutes with extended tensor products of finite sets with group actions and how the Brauer construction commutes with taking extended tensor products of $p$-permutation modules. These fundamental formulas, generalizing earlier results by Boltje-Danz and by Boltje-Perepelitsky, should be of independent interest.