On the Evaluation of the Multiplicity-Free Wigner Coefficients of U(n)

Abstract

An explicit algebraic expression [containing the minimal number (n − 1) sums] for the general reduced Wigner coefficients associated with the multiplicity free Kronecker product [h1 … hn] × [p0 … 0] of irreducible representations of U(n), is determined. The calculation employs a combined use of recursive relations derived for the Wigner coefficients, and matrix elements (with respect to Gel'fand basis states) of a generator of U(n) raised to an arbitrary power. We also give an alternative procedure using the techniques of the ``pattern calculus.'' The method is illustrated first for the case of U(3) and then generalized to arbitrary U(n). It is found that the results can be expressed succinctly in terms of a new algebraic function Snm - thereby elucidating the structural details of the underlying algebra.