The Bruhat Order of a Finite Coxeter Group and Elnitsky Tilings

TL;DR

This paper establishes a constructive link between the Bruhat order of finite Coxeter groups and Elnitsky tilings, providing algorithms and examples including the exceptional group E8.

Contribution

It introduces a method to produce Elnitsky tilings for any element in a finite Coxeter group based on reduced expressions, unifying previous concepts and providing computational tools.

Findings

Constructive proof connecting Bruhat order and Elnitsky tilings.

Algorithms for generating tilings for various Coxeter groups.

Examples including the exceptional group E8.

Abstract

Suppose that $W$ is a finite Coxeter group and $W_J$ a standard parabolic subgroup of $W$. The main result proved here is that for any for any $w \in W$ and reduced expression of $w$ there is an Elnitsky tiling of a $2m$-polygon, where $m = [W : W_J]$. The proof is constructive and draws together the work on E-embedding in \cite{nicolaidesrowley1} and the deletion order in \cite{nicolaidesrowley3}. Computer programs which produce such tilings may be downloaded from \cite{github} and here we also present examples of the tilings for, among other Coxeter groups, the exceptional Coxeter group $\mathrm{E}_8$.