The symmetric group and the unitary group : an application of group-subgroup transformation theory

Abstract

The general mathematical aspects of transformation theory of an arbitrary group-subgroup scheme are developed. The theorems of the Racah-Wigner coupling algebra are shown to be a special case of this general theory. Several new transformation factors are defined, and symmetries and a calculational method based on the Butler method are presented. The Butler method is used to obtain n dependent algebraic formulae for some 6j symbols of the special unitary group SUn. Combined with the composite labelling for SUn irreps, several symmetries of these 6j formulae n are apparent, in particular the transpose conjugate symmetry of the symmetric group. The close relationship between the unitary group and the symmetric group are reviewed and many relations developed. In particular duality factors are defined and their symmetry and phase freedom properties are discussed. It is found that in connection with the Un≃U₁xSUn isomorphism, insufficient phase freedom exists to choose all duality and isomorphism factors.