Phase-fixed double-group 3-Γ symbols. V. 3-Γ symbols and coupling coefficients for all the octahedral double group-subgroup hierarchies |
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| Authors: | Ture Damhus Sven E Harnung Claus E Schäffer |
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| Institution: | (1) Chemistry Department I, H. C. Ørsted Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark;(2) Present address: Department of Pharmaceutical chemistry AD, Royal Danish School of Pharmacy, Universitetsparken 2, DK-2100 Copenhagen Ø, Denmark |
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| Abstract: | For all the subgroup hierarchies descending from the octahedral double group O
*, we have obtained sets of 3- symbols and discuss here their properties. We have entirely real sets of 3- symbols for the tetrahedral and tetragonal hierarchies as well as for O
*  C
3
*
. For the latter hierarchy and the tetragonal ones, formalisms almost as powerful as the classical one for the rotation group may be established. We also discuss results obtained for cases with strict adaption to D
3
*
where it is now known that non-real 3- symbols are unavoidable.The 3- symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them.The significance of the concept of associated representations of O* is discussed. The problems raised by the two multiplicity triples UT1U and UT2U in O* are given particular attention. |
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| Keywords: | octahedral double group-subgroup hierarchies real phase-fixed three-gamma symbols and coupling coefficients standard irreducible matrix representations complex conjugation of matrix representations by inner automorphism non-real trigonally adapted octahedral three-gamma symbols |
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