Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Abstract

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

Geometrical structures of nested polyhedra

“…In the case of a non-crystallographic group H 3 , there is no corresponding Lie algebra. However, procedure for finding lower orbits can be repeated with some modifications to the algorithm [11]. The procedure includes the following steps:…”