The rings of n-dimensional polytopes

Details The rings of n-dimensional polytopes The rings of n-dimensional polytopes

2009-01-29

arxiv.org/abs/0901.4686v1

L. Hakova, M. Larouche, J. Patera,The rings of n-dimensional polytopes. J. Phys. A: Math. Theor. 41 (2008) 495202 (21pp)

Mathematics

Metric Geometry

24 pages

Scientific paper

10.1088/1751-8113/41/49/495202

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.

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