Phase conventions consistent with duality between SUn and S L

Abstract

The Schur–Weyl construction of irreducible tensors by symmetrized Lth rank exterior products of a defining n-dimensional vector space establishes a duality between the coupling algebra of the unitary group Un and the symmetric group SL . The coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in Un and association in SL . The unitary group is factored by its unitary unimodular subgroup SUn≂Un/U1 so it is usual to identify the coupling algebras of Un and SUn by choosing the trivial (phaseless) coupling for U1. This establishes a set of equivalence relations since all pseudoscalar irreducible representations (irreps) of Un subduce onto the scalar irrep of SUn. The question remains can this trivial identification with the coupling algebra of SUn be made consistent with the symmetries which follow from duality? Specifically the symmetries which follow from duality must be consistent with the equivalence relations. We argue such consistency may be obtained. The resulting symmetries place significant restrictions on the resolution of multiplicities and the choice of phase conventions for transformations which in the absence of duality are normally considered independently. We review the usual formulation of a coupling algebra and the identifications which follow from duality using a double coset decomposition of SL . Particular attention is given to the phase transformation introduced by a transposition in order of the two component irreps being coupled. Phase transformations needed for association and complex conjugation are examined to determine the symmetries they must exhibit to be consistent with duality. Two phase transformations needed for complex conjugation, the Derome–Sharp matrix and the 1jm factor, are shown to be related by association. A transposition phase convention and an association phase convention are proposed and shown to be consistent with all the symmetries required by duality.