Multiplicity-free Wigner coefficients for semisimple Lie groups. I. The U(n) pattern calculus

This is the first paper in a series of two dedicated to a new algebraic determination of the multiplicity-free reduced Wigner coefficients for the Lie groups U(n) and O(n). The approach employed enables a direct (nonrecursive) derivation of reduced Wigner coefficients. The absolute value squared of the reduced Wigner coefficients is expressed as a rational polynomial function (numerator polynomial divided by denominator polynomial) whose Weyl group symmetries are seen to fall out naturally in our approach from the transformation properties of polynomial functions determined by Casimir invariants. A unified treatment of the composition laws of reduced Wigner coefficients and the evaluation of their associated denominator polynomials is presented, which applies to both U(n) and O(n). An explicit formula for the numerator polynomials of U(n) is also derived. The numerator polynomials for the orthogonal groups will be given in the second paper of the series.