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.