The Bruhat Order of a Finite Coxeter Group and Elnitsky Tilings

2024/07/10 by Robert Nicolaides, Nicolaides, Robert, Peter Rowley +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2407.07975

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 ∈ 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 \citenicolaidesrowley1 and the deletion order in \citenicolaidesrowley3. Computer programs which produce such tilings may be downloaded from \citegithub and here we also present examples of the tilings for, among other Coxeter groups, the exceptional Coxeter group E₈.

Related

• The Deletion Order and Coxeter Groups
• Finite Coxeter Groups and Generalized Elnitsky Tilings
• Subtilings of Elnitsky Tilings for Finite Irreducible Coxeter Groups
• A note on Bruhat order and double coset representatives
• Weak Bruhat order on the set of faces of the permutahedra