Diameters of graphs of reduced words and rank-two root subsystems

TL;DR

This paper investigates the diameter of graphs formed by reduced words in Coxeter groups, proving a tight lower bound for symmetric groups and exploring cases in other classical types.

Contribution

It resolves conjectures on the diameter of these graphs for symmetric groups and characterizes cases of equality, advancing understanding in Coxeter group combinatorics.

Findings

Proved a tight lower bound for the diameter in symmetric groups.

Characterized equality cases in the diameter bounds.

Provided partial results for classical types beyond symmetric groups.

Abstract

We study the diameter of the graph $G(w)$ of reduced words of an element $w$ in a Coxeter group $W$ whose edges correspond to applications of the Coxeter relations. We resolve conjectures of Reiner--Roichman and Dahlberg--Kim by proving a tight lower bound on this diameter when $W=S_n$ is the symmetric group and by characterizing the equality cases. We also give partial results in other classical types which illustrate the limits of current techniques.