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The invariants of finite-dimensional representations of simple Lie algebras,such as even-degree indices and anomaly numbers, are considered in the contextof the non-crystallographic finite reflection groups $H_2$, $H_3$ and $H_4$.Using a representation-orbit replacement, the definitions and properties of theindices are formulated for individual orbits of the examined groups. Theindices of orders two and four of the tensor product of $k$ orbits aredetermined. Using the branching rules for the non-crystallographic Coxetergroups, the embedding index is defined similarly to the Dynkin index of arepresentation. Moreover, since the definition of the indices can be applied toany orbit of non-crystallographic type, the algorithm allowing to search forthe orbits of smaller radii contained within any considered one is presentedfor the Coxeter groups $H_2$ and $H_3$. The geometrical structures of nestedpolytopes are exemplified.

Nested polyhedra and indices of orbits of Coxeter groups of non-crystallographic type