Faces of root polytopes in all dimensions

Abstract

In this paper the root polytopes of all finite reflection groupsWwith 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 appropriateW-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter–Dynkin diagram [Champagneet 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.