Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type
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>&...
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MDPI AG
2020-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/10/1737 |
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| author | Mariia Myronova Jiří Patera Marzena Szajewska |
| author_facet | Mariia Myronova Jiří Patera Marzena Szajewska |
| author_sort | Mariia Myronova |
| collection | DOAJ |
| description | 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. |
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| last_indexed | 2024-03-10T15:29:06Z |
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| series | Symmetry |
| spelling | doaj.art-f3f4b1a7ccb24d2998b38cdf085b80c22023-11-20T17:49:41ZengMDPI AGSymmetry2073-89942020-10-011210173710.3390/sym12101737Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic TypeMariia Myronova0Jiří Patera1Marzena Szajewska2Département de Physique, Université de Montréal, Complexe des Sciences, 1375 Avenue Thérèse-Lavoie-Roux, Montréal, QC H2V 0B3, CanadaCentre de Recherches Mathématique, Université de Montréal, C. P. 6128 Centre-Ville, Montréal, QC H3C 3J7, CanadaDepartment of Mathematics, University of Białystok, 1M Ciołkowskiego, PL-15-245 Białystok, PolandThe 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.https://www.mdpi.com/2073-8994/12/10/1737Coxeter groupnested polytopeorbit indexhigher-order indexanomaly numberweight multiplicity |
| spellingShingle | Mariia Myronova Jiří Patera Marzena Szajewska Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type Symmetry Coxeter group nested polytope orbit index higher-order index anomaly number weight multiplicity |
| title | Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type |
| title_full | Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type |
| title_fullStr | Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type |
| title_full_unstemmed | Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type |
| title_short | Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type |
| title_sort | nested polyhedra and indices of orbits of coxeter groups of non crystallographic type |
| topic | Coxeter group nested polytope orbit index higher-order index anomaly number weight multiplicity |
| url | https://www.mdpi.com/2073-8994/12/10/1737 |
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