Faces of root polytopes in all dimensions

Abstract

In this paper the root polytopes of all finite reflection groups W with a connected Coxeter-Dynkin diagram in {\bb R}^n are identified, their faces of dimensions 0 ≤ d ≤ n - 1 are counted, and the construction of representatives of the appropriate W-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter-Dynkin diagram [Champagne et al. (1995). Can. J. Phys. 73, 566-584]. Each recursion step provides the essentials of faces of a specific dimension and specific symmetry. The results can be applied to crystals of any dimension and any symmetry.

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

“…It was first introduced by Moody & Patera (1992). In recent years, the method has been used, for example, to describe the faces of Platonic solids (Szajewska, 2014) and root polytopes (Szajewska, 2016) in all dimensions.…”