Theoretica chimica acta

, Volume 65, Issue 5–6, pp 317–363 | Cite as

Phase-fixed double-group 3-Γ symbols. I. A novel exposition of the general theory of 3-Γ symbols and coupling coefficients

  • Ture Damhus
  • Sven E. Harnung
  • Claus E. Schäffer
Original Investigations


The present paper is the first in a series aiming at the establishment of a transparent and readily applicable Wigner-Racah algebra for all the noncommutative double groups.

Starting from the Wigner-Eckart theorem in a very general setting, the theory of the fundamental quantities called here triple coefficients — and the closely related coupling coefficients — is developed and leads through a careful discussion of permutational symmetries to the concept of 3symbols. By basing the exposition on triple coefficients and by consistently using matrix representations, we obtain a notation and a terminology which enable a clear separation of permutational properties and problems concerning complex conjugation, and a more transparent discussion of tensor (Kronecker) product multiplicities.

A particularly elegant formalism is obtained for a situation which generalizes that of the classical rotation-group Wigner-Racah algebra, viz., in which there is a fixed group element effecting (through the inner automorphism it defines) complex conjugation of all the standard irreducible matrix representations.

Key words

three-gamma symbols coupling coefficients triple coefficients complex conjugation of irreducible matrix representations inner automorphisms Frobenius-Schur classification Derome-Sharp matrices Wigner-Racah algebra 


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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Ture Damhus
    • 1
  • Sven E. Harnung
    • 1
  • Claus E. Schäffer
    • 1
  1. 1.Chemistry Department I, H. C. Ørsted InstituteUniversity of CopenhagenCopenhagen ØDenmark

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