Commuting categories for blocks and fusion systems

TL;DR

This paper generalizes the concept of commuting posets and graphs from finite groups to p-blocks and fusion systems, providing a simplified proof of a known result using G-equivariant methods.

Contribution

It extends the notion of commuting structures to blocks and fusion systems and offers a concise proof of a key result in this context.

Findings

Generalization of commuting posets to blocks and fusion systems

Simplified proof of a result using G-equivariant methods

Extension of Alperin's and Quillen's results to new settings

Abstract

We extend the notion of a commuting poset for a finite group to p-blocks and fusion systems, and we generalize a result, due originally to Alperin and proved independently by Aschbacher and Segev, to commuting graphs of blocks, with a very short proof based on the G-equivariant version, due to Thevenaz and Webb, of a result of Quillen.