Complementary group resolution of the $SU(n)$ outer multiplicity problem. I. The Littlewood rules and a complementary $U(2n−2)$ group structure

A complementary group to $SU(n)$ is found that realizes all features of the Littlewood rules for Kronecker products of $SU(n)$ representations. This is accomplished by considering a state of $SU(n)$ to be a special Gel’fand state of the complementary group $U(2n−2)$ with labels of the latter used to distinguish multiple occurrences of irreducible representations of $SU(n)$ (irreps) in the $SU(n)×SU(n)↓SU(n)$ decomposition that is obtained from the Littlewood rules. Furthermore, this realization also helps us to determine $SU(n)⊃SU(n−1)×U(1)$ Reduced Wigner Coefficients (RWCs, frequently called Isoscalar Factors) and Clebsch–Gordan Coefficients [CGCs, or full (nonreduced) Wigner Coefficients] of $SU(n),$ using algebraic or numeric methods, in either the canonical or a noncanonical basis. New explicit formulas for the SU(3) and SU(4) multiplicities are obtained by using this technique.