Finite Coxeter Groups and Generalized Elnitsky Tilings

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