Some combinatorial models for reduced expressions in Coxeter groups

TL;DR

This paper develops a combinatorial framework to enumerate and analyze reduced expressions in arbitrary Coxeter groups, extending known results and characterizations to a broader class of groups.

Contribution

It introduces a universal combinatorial model for reduced expressions and provides a new characterization of elements with specific deletion properties across all Coxeter groups.

Findings

Framework for representing inversion sets and reduced expressions

A formula for element length after generator deletion

Generalization of properties of freely braided elements

Abstract

Stanley's formula for the number of reduced expressions of a permutation regarded as a Coxeter group element raises the question of how to enumerate the reduced expressions of an arbitrary Coxeter group element. We provide a framework for answering this question by constructing combinatorial objects that represent the inversion set and the reduced expressions for an arbitrary Coxeter group element. The framework also provides a formula for the length of an element formed by deleting a generator from a Coxeter group element. Fan and Hagiwara, et al$.$ showed that for certain Coxeter groups, the short-braid avoiding elements characterize those elements that give reduced expressions when any generator is deleted from a reduced expression. We provide a characterization that holds in all Coxeter groups. Lastly, we give applications to the freely braided elements introduced by Green and Losonczy, generalizing some of their results that hold in simply-laced Coxeter groups to the arbitrary Coxeter group setting.