The rings of n-dimensional polytopes

Abstract

Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.

24 pages

References in corpus (2)

• New group structures for Carbon onions and Carbon nanotubes via affine extensions of non-crystallographic Coxeter groups
• Fourier transform method for imaging atmospheric Cherenkov telescopes

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• On Discretization of Tori of Compact Simple Lie Groups
• Complete Decompositions of Coxeter groups orbit products of $A_2,C_2,G_2$ and $H_2$ and some limits for them