Cycles in subexpression graphs

TL;DR

This paper investigates the structure of subexpression graphs in Coxeter groups, proving their connectivity and characterizing their cycle spaces in terms of divisors of finite orders of element products.

Contribution

It establishes the connectivity of subexpression graphs and describes their cycle space structure in terms of divisors of finite orders, a novel structural insight.

Findings

Subexpression graphs are connected.

Cycle space is spanned by cycles of specific lengths.

Cycle lengths relate to divisors of finite orders.

Abstract

Let $\mathfrak{S}(\underline{s},w)$ be the graph whose vertices are all subexpressions with target $w$ of a fixed expression $\underline{s}$ in generators of a Coxeter group and edges are the pairs of subexpressions with Hamming distance 2. We prove that $\mathfrak{S}(\underline{s},w)$ is connected and its cycle space is spanned by cycles of lengths $d+2$, where $d$ ranges over all positive divisors of all finite orders of products of at most two entries of $\underline{s}$.