A Note on Commuting Involution Graphs in Affine Coxeter Groups

TL;DR

This paper investigates the structure of commuting involution graphs within affine Coxeter groups, establishing general results and addressing specific types, thereby extending understanding beyond classical cases.

Contribution

It provides new general results for commuting involution graphs in affine Coxeter groups and analyzes types F_4 and G_2, with plans to study more complex types in future work.

Findings

Established general results for affine Coxeter groups

Analyzed commuting involution graphs for types F_4 and G_2

Outlined future work for types E_6, E_7, E_8

Abstract

Commuting involution graphs have been studied for finite Coxeter groups and for affine groups of classical type. The purpose of this short note is to establish some general results for commuting involution graphs in affine Coxeter groups, and to deal with types $\tilde F_4$ and $\tilde G_2$. Types $\tilde E_6$ $\tilde E_7$ and $\tilde E_8$ are more substantial and we will address these in a forthcoming paper.