Multiplicity-free 6-j symbols and Weyl coefficients of U(n): Explicit evaluation

Abstract

An explicit expression has been obtained for all known multiplicity‐free 6‐j symbols of U(n), i.e., 6‐j symbols of the following three types, where anyone of its columns consists of (1) two totally symmetric representations, (2) one totally symmetric and one conjugate to the totally symmetric, and (3) two conjugate to the totally symmetric. The symmetry properties of the multiplicity‐free 6‐j symbols of U(n) under permutation of columns, inversion of any two columns, and conjugation are given. Some general theorems concerning the multiplicity‐free 6‐j symbols of U(n) or more precisely, the multiplicity‐free 6‐j symbols of the ’’SU(n) type’’ have been obtained. Since the Weyl coefficients of U(n) are basically 6‐j symbols of U(n−1), we also conclude that the Weyl coefficients of U(n) have been explicitly obtained. This result implies that the d function of U(n) can be completely and explicitly written down in terms of the Weyl coefficients.