Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability

TL;DR

This paper develops a framework for relative stable equivalences of Morita type for principal blocks of finite groups sharing a central p-subgroup, extending existing theories and introducing the concept of relative Brauer indecomposability.

Contribution

It introduces a method to construct relative Z-stable Morita equivalences and generalizes prior results on stable equivalences of Morita type to this new setting.

Findings

Constructed a method for relative Z-stable Morita equivalences

Generalized Linckelmann's results to the relative setting

Provided an equivalent condition for Scott modules to be relatively Brauer indecomposable

Abstract

We discuss representations of finite groups having a common central $p$-subgroup $Z$, where $p$ is a prime number. For the principal $p$-blocks, we give a method of constructing a relative $Z$-stable equivalence of Morita type, which is a generalization of a stable equivalence of Morita type, and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann's results on stable equivalences of Morita type to relative $Z$-stable equivalences of Morita type. We also introduce the notion of relative Brauer indecomposability, which is a generalization of the notion of Brauer indecomposability. We give an equivalent condition for Scott modules to be relatively Brauer indecomposable, which is an analogue of that given by Ishioka and the first author.