Origin of regge symmetry for wigner coefficients ofLU(2)

Using general properties of the representations of unitary groups and their relations to representations of symmetric groups, the 3j symbol of the unitary unimodular group ?U(2) is written in terms of a 9j symbol of the unitary unimodular group ?U(J) withJ being the sum of the threej's. The result yields the Regge symmetry of the 3j symbol as a consequence of new relations between Wigner coefficients and special invariants of unitary groups on one hand and the association symmetry of the symmetric group on the other.