Proof of a conjecture of Bergeron, Ceballos and Labb\'e

TL;DR

This paper proves a bipartiteness-type property of a graph formed by reduced expressions in Coxeter groups, strengthening a previous result and exploring braid move structures.

Contribution

It establishes a strong bipartiteness property for the graph of reduced expressions, generalizing and strengthening a prior 2014 result by Bergeron, Ceballos, and Labbé.

Findings

Every cycle in the graph has even length.

Arcs can be colored with pairs of opposite colors.

In any cycle, arcs of each color pair occur an even number of times.

Abstract

The reduced expressions for a given element $w$ of a Coxeter group $(W, S)$ can be regarded as the vertices of a directed graph $\mathcal{R}(w)$; its arcs correspond to the braid moves. Specifically, an arc goes from a reduced expression $a$ to a reduced expression $b$ when $b$ is obtained from $a$ by replacing a contiguous subword of the form $stst...$ (for some distinct $s, t$ in $S$) by $tsts...$ (where both subwords have length $m_{s, t}$, the order of $st$ in $W$). We prove a strong bipartiteness-type result for this graph $\mathcal{R}(w)$: Not only does every cycle of $\mathcal{R}(w)$ have even length; actually, the arcs of $\mathcal{R}(w)$ can be colored (with colors corresponding to the type of braid moves used), and to every color $c$ corresponds an "opposite" color $c^{\operatorname{op}}$ (corresponding to the reverses of the braid moves with color $c$), and for any color $c$, the number of arcs in any given cycle of $\mathcal{R}(w)$ having color in $\left\{c, c^{\operatorname{op}}\right\}$ is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labb\'e. We state further conjectural extensions.