Saturated fusion systems with parabolic families

TL;DR

This paper explores the relationship between fusion systems and chamber systems in group theory, introducing parabolic families and conditions for saturation, with applications to classical types.

Contribution

It introduces the concept of fusion systems with parabolic families and links them to chamber systems, providing conditions for saturation and applications to classical types.

Findings

Established conditions for chamber systems to ensure fusion system saturation

Linked fusion systems with parabolic families to chamber system structures

Applied theory to classical type fusion systems

Abstract

Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N_G(S). Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.