Isotypic blocks of finite groups algebras that are not $p$-permutation equivalent

TL;DR

This paper investigates the relationship between isotypy and p-permutation equivalence in finite group algebra blocks, showing that isotypy does not always imply p-permutation equivalence through counterexamples.

Contribution

It demonstrates that Galois conjugate blocks can be isotypic without being p-permutation equivalent, clarifying the distinction between these concepts.

Findings

Counterexamples of Galois conjugate blocks that are isotypic but not p-permutation equivalent

Analysis of conditions under which isotypy does not lift to p-permutation equivalence

Clarification of the relationship between isotypy and p-permutation equivalence in block theory

Abstract

We show that Kessar's isotypy between Galois conjugate blocks of finite group algebras does not always lift to a $p$-permutation equivalence. We also provide examples of Galois conjugate blocks which are isotypic but not $p$-permutation equivalent. These results help to clarify the distinction between a $p$-permutation equivalence and an isotypy, and may be useful in determining necessary and sufficient conditions for when an isotypy lifts to a $p$-permutation equivalence.