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

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Département de Physique, Université de Montréal, Complexe des Sciences, 1375 Avenue Thérèse-Lavoie-Roux, Montréal, QC H2V 0B3, Canada
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Centre de Recherches Mathématique, Université de Montréal, C. P. 6128 Centre-Ville, Montréal, QC H3C 3J7, Canada
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Department of Mathematics, University of Białystok, 1M Ciołkowskiego, PL-15-245 Białystok, Poland
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(10), 1737; https://doi.org/10.3390/sym12101737
Received: 14 August 2020 / Revised: 29 September 2020 / Accepted: 13 October 2020 / Published: 20 October 2020
(This article belongs to the Special Issue Symmetry in Discrete and Combinatorial Geometry)
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. View Full-Text
Keywords: Coxeter group; nested polytope; orbit index; higher-order index; anomaly number; weight multiplicity; search algorithm; tree-diagram Coxeter group; nested polytope; orbit index; higher-order index; anomaly number; weight multiplicity; search algorithm; tree-diagram
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Myronova, M.; Patera, J.; Szajewska, M. Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type. Symmetry 2020, 12, 1737.

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