Recursion Relations for the Wigner Coefficients of Unitary Groups

Abstract

The polynomials in the components of a set of n-dimensional vectors that form a basis for an irreducible representation of S𝒰n are shown to be part of the basis of the group Unr, in which the subgroup 𝒰n × Ur is explicitly reduced and r ≥ n - 1. Using this result, the concept of auxiliary Wigner coefficient is introduced, for which the problem of multiplicity does not arise and the phase convention is related to that of Gel'fand and Zetlin; recursion relations for this auxiliary coefficient are obtained in a straightforward way, and the connection between it and the ordinary Wigner coefficient is shown to be simple. The recursion relations are being programmed for an electronic computer to allow the systematic evaluation of the Wigner coefficients of S𝒰3 and S𝒰4.