On Morita equivalences with endopermutation source and isotypies

TL;DR

This paper introduces the concept of almost isotypies as a new, weaker form of equivalence between blocks of finite group algebras, extending Morita equivalences with endopermutation sources.

Contribution

It defines almost isotypies, shows their relation to Morita equivalences with endopermutation sources, and establishes conditions under which these equivalences induce almost isotypies.

Findings

Morita equivalence with endopermutation source implies almost isotypy.

Almost isotypies are a weaker form of Broué's isotypy.

Rational integer character values of sources are key to inducing almost isotypies.

Abstract

We introduce a new type of equivalence between blocks of finite group algebras called an almost isotypy. An almost isotypy restricts to a weak isotypy in Brou\'{e}'s original definition, and it is slightly weaker than Linckelmann's version. We show that a bimodule of two block algebras of finite groups - which has an endopermutation module as a source and which induces a Morita equivalence - gives rise, via slash functors, to an almost isotypy if the character values of a (hence any) source are rational integers. Consequently, if two blocks are Morita equivalent via a bimodule with endopermutation source, then they are almost isotypic. We also explain why the notion of almost isotypies is reasonable.