Brauer-friendly modules and slash functors

TL;DR

This paper defines Brauer-friendly modules as a generalization of endo-p-permutation modules, introduces a functorial slash construction, and applies these tools to classify modules and establish stable equivalences in block theory.

Contribution

It introduces Brauer-friendly modules and a functorial slash construction, extending Dade's slash functor, enabling classification and equivalence results in block theory.

Findings

Defined Brauer-friendly modules as a generalization of endo-p-permutation modules.

Developed a functorial version of Dade's slash construction for these modules.

Proved the existence of stable equivalences between certain blocks.

Abstract

This paper introduces the notion of Brauer-friendly modules, a generalisation of endo-p-permutation modules. A module over a block algebra OGe is said to be Brauer-friendly if it is a direct sum of indecomposable modules with compatible fusion-stable endopermutation sources. We obtain, for these modules, a functorial version of Dade's slash construction, also known as deflation-restriction. We prove that our slash functors, defined over Brauer-friendly categories, share most of the very useful properties that are satisfied by the Brauer functor over the category of p-permutation OGe-modules. In particular, we give a parametrisation of indecomposable Brauer-friendly modules, which opens the way to a complete classification whenever the fusion-stable sources are classified. Those tools have been used to prove the existence of a stable equivalence between non-principal blocks in the context of a minimal counter-example to the odd Z*p-theorem.