Commuting Involution Graphs in Classical Affine Weyl Groups

TL;DR

This paper studies the structure of commuting involution graphs in classical affine Weyl groups, establishing bounds on their connectivity and diameter for different types, extending known results to new cases.

Contribution

It proves that for classical affine Weyl groups, the commuting involution graphs are either disconnected or have diameter at most n+2, including new proofs for types B and D.

Findings

The commuting involution graph is either disconnected or has diameter ≤ n+2.

The bound is confirmed for types A_n and C_n.

New proofs are provided for types B_n and D_n.

Abstract

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let $W$ be a classical Weyl group of rank $n$, with $\tilde W$ its corresponding affine Weyl group. Our main result is that if $X$ is a conjugacy class of involutions in $\tilde W$, then the commuting involution graph $\mathcal{C}(\tilde W, X)$ is either disconnected or has diameter at most $n+2$. This bound is known to hold for types $\tilde A_n$ and $\tilde C_n$, so the main work of this paper is to prove the theorem for types $\tilde B_n$ and $\tilde D_n$.