The Bruhat Order of a Finite Coxeter Group and Elnitsky Tilings

Suppose that W is a finite Coxeter group and WJ a standard parabolic subgroup of W. The main result proved here is that for any for any w ∈ W and reduced expression of w there is an Elnitsky tiling of a $2m$-polygon, where m = [W : WJ]. The proof is constructive and draws together the work on E-embedding in {nicolaidesrowley1} and the deletion order in {nicolaidesrowley3}. Computer programs which produce such tilings may be downloaded from {github} and here we also present examples of the tilings for, among other Coxeter groups, the exceptional Coxeter group E₈.