On the structure of the multiplicity-free Wigner coefficients of U(n)

Abstract

It is shown that the isoscalar factor (or reduced Wigner coefficient) in U(n) is essentially a doubly stretched 9‐j symbol in U(n−1). The connection between the isoscalar factorand the 9‐j symbol of U(n) and U(n−1) is also noted. This result immediately implies that the Weyl coefficients of U(n) are basically 6‐j symbols of U(n−1), a result first noted by Holman. The finite transformation matrix D^[m]n_{(m′)n−1(m)n−1} either in terms of generalized Euler angles or double bosons can thus be written down in a simple way. The stretched 6‐j symbols of U(n) are obtained in a simple form, involving no summations. The generalized beta functions of Gel’fand and Graev for U(n) are found to be connected with the stretched 6‐j symbols of U(n−1) and an isoscalar factor of U(n−1). The 144 Regge symmetries of the 6‐j symbol of U(2) can be interpreted as the symmetries of the Weyl coefficients of the double boson state of U(3) ‐U(3). In the Appendix we give the phase relations between the Wigner coefficients and 3‐j symbols of U(n), a result which is by no means trivial, and is of some practical importance.