Braid graphs in simply-laced triangle-free Coxeter systems are median

TL;DR

This paper proves that braid graphs in simply-laced triangle-free Coxeter systems are median graphs, providing an alternative proof and determining the minimal hypercube dimension for embedding these graphs.

Contribution

It offers an alternative proof that braid graphs are median and identifies the minimal hypercube embedding dimension, addressing an open question.

Findings

Braid graphs are median in simply-laced triangle-free Coxeter systems.

The minimal hypercube dimension for embedding braid graphs is determined.

Provides an alternative proof to previous results.

Abstract

Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. Two reduced expressions are said to be braid equivalent if they are related via a sequence of braid moves. Braid equivalence is an equivalence relation and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In a recent paper, Awik et al.~proved that when a Coxeter system is simply laced and triangle free (i.e., the corresponding Coxeter graph has no three-cycles), the braid graph for a reduced expression is a partial cube (i.e., isometric to a subgraph of a hypercube). In this paper, we will provide an alternate proof of this fact, as well as determine the minimal dimension hypercube into which a braid graph can be isometrically embedded, which addresses an open question posed by Awik et al. For our main result, we prove that braid graphs in simply-laced triangle-free Coxeter systems are median, which is a strengthening of previous results.