Fusion systems of blocks of finite groups over arbitrary fields

TL;DR

This paper explores the structure of fusion systems associated with blocks of finite groups over arbitrary fields, especially in non-split cases, providing explicit descriptions and conditions for saturation.

Contribution

It extends Puig's theory by describing how fusion systems in non-split cases relate to larger fields and automorphisms, clarifying their structure.

Findings

Fusion systems in non-split cases can be explicitly described.

Fusion systems are generated by those over larger fields and a single automorphism.

Conditions for saturation are characterized in the non-split setting.

Abstract

To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a maximal $kGb$-Brauer pair. We investigate in the non-split case how far the fusion system is from being saturated by describing it in an explicit way as being generated by the fusion system of a related block idempotent over a larger field together with a single automorphism of the defect group.