The Bruhat Order of a Finite Coxeter Group and Elnitsky Tilings

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 ${\mathrm{E}}_8$.