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

• Main Authors: Mariia Myronova, Jiří Patera, Marzena Szajewska
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Symmetry
• Subjects: Coxeter group; nested polytope; orbit index; higher-order index; anomaly number; weight multiplicity
• Online Access: https://www.mdpi.com/2073-8994/12/10/1737

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 <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>3</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>4</mn></msub></semantics></math></inline-formula>. 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 <i>k</i> 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 <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mn>3</mn></msub></semantics></math></inline-formula>. The geometrical structures of nested polytopes are exemplified.