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

Abstract

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. © 1998 American Institute oj Physics.