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 theLittlewood rule for Kronecker products of SU(n) representations. This isaccomplished by considering a state of SU(n) to be a special Gel'fand state ofthe complementary group {\cal U}(2n-2). The labels of {\cal U}(2n-2) can beused as the outer multiplicity labels needed to distinguish multipleoccurrences of irreducible representations (irreps) in the SU(n)\timesSU(n)\downarrow SU(n) decomposition that is obtained from the Littlewood rule.Furthermore, this realization can be used to determine SU(n)\supsetSU(n-1)\times U(1) Reduced Wigner Coefficients (RWCs) and Clebsch-GordanCoefficients (CGCs) of SU(n), using algebraic or numeric methods, in either thecanonical or a noncanonical basis. The method is recursive in that it usessimpler RWCs or CGCs with one symmetric irrep in conjunction with standardrecoupling procedures. New explicit formulae for the multiplicity for SU(3) andSU(4) are used to illustrate the theory.Comment: 15 pages, LaTe