Local recognition of reflection graphs on Coxeter groups

TL;DR

This paper investigates how reflection graphs associated with Coxeter groups can be recognized locally, especially for types F4 and An, and explores their implications for classifying finite simple groups.

Contribution

It provides new local recognition results for reflection graphs on Coxeter groups, particularly F4, and links these results to the recognition of Chevalley groups.

Findings

Reflection graphs on Coxeter groups can be locally recognized under certain conditions.

Recognition results help characterize Coxeter groups via reflection centralizers.

Connections to Chevalley groups aid in classifying finite simple groups.

Abstract

We provide local recognition results for the reflection graphs on spherical Coxeter groups. In particular, we study the case $F_4$ which is locally recognizable under additional constraints only. It is then demonstrated in the cases $A_n$ and $F_4$ how these graph theoretical recognition results can be used to characterize the corresponding Coxeter groups in terms of their reflection centralizers. Finally, we outline the connection to the local recognition of Chevalley groups which is particularly important in the classification of the finite simple groups.