Finite Coxeter Groups and Generalized Elnitsky Tilings

TL;DR

This paper generalizes Elnitsky's bijections between reduced words of Coxeter groups and polygon tilings from types A, B, D to all finite Coxeter groups using embeddings into the symmetric group.

Contribution

It extends Elnitsky's tiling correspondence to all finite Coxeter groups through a new embedding approach.

Findings

Unified tiling framework for all finite Coxeter groups

New bijections between reduced words and polygon tilings

Enhanced understanding of Coxeter group structures

Abstract

In [5], Elnitsky constructed three elegant bijections between classes of reduced words for Type $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{D}$ families of Coxeter groups and certain tilings of polygons. This paper offers a particular generalization of this concept to all finite Coxeter Groups in terms of embeddings into the Symmetric Group. [5] Elnitsky, Serge. Rhombic tilings of polygons and classes of reduced words in Coxeter groups. PhD dissertation, University of Michigan, 1993.