The rings of n-dimensional polytopes

TL;DR

This paper explores the geometric and algebraic properties of polytopes generated by finite Coxeter groups, introducing methods for their description, decomposition, and invariants, applicable to both crystallographic and non-crystallographic groups.

Contribution

It provides a comprehensive framework for describing G-polytopes, their products, decompositions, and invariants, extending previous work to a broader class of Coxeter groups.

Findings

Introduced efficient geometric description methods for G-polytopes

Defined invariants such as Dynkin index analogs and anomaly numbers

Applied concepts to both crystallographic and non-crystallographic Coxeter groups

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.