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Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type. Symmetry (Basel) 2020. [DOI: 10.3390/sym12101737] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
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.
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Okubo S, Tosa Y. Further study of global gauge anomalies of simple groups. Int J Clin Exp Med 1989; 40:1925-1933. [PMID: 10012021 DOI: 10.1103/physrevd.40.1925] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Tosa Y. Global gauge anomalies for theories with the Green-Schwarz local-anomaly-cancellation mechanism. Int J Clin Exp Med 1989; 40:1934-1937. [PMID: 10012022 DOI: 10.1103/physrevd.40.1934] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Zhang H, Okubo S. Global gauge anomaly and James numbers of Stiefel manifolds. Int J Clin Exp Med 1988; 38:1880-1887. [PMID: 9959339 DOI: 10.1103/physrevd.38.1880] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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