Abstract
The state labelling problem arising in the reduction of the direct product of a p positive-row U(n) irreducible representation (h1. . .hp0) with a q negative-row one (0-h'q. . .-h'1) into a sum of mixed U(n) irreducible representations (k1. . .kp0-k'q. . .-k'1) is solved by using the complementarity between U(n) and U(p,q) within some positive discrete series irreducible representations of U(pn,qn). This complementarity allows analysis of the problem in terms of the group chain U(p,q) contains/implies U(p)*U(q) instead of U(n)*U(n) contains/implies U(n). For the most general SU(n) irreducible representations corresponding to p=q=n-1, the relevant group chain is therefore U(n-1,n-1) contains/implies U(n-1)*U(n-1). In such a case, the additional labels include those of an intermediate U(n-1) irreducible representation (h1s. . .hn-1s), as well as the additional labels solving the state labelling problems for the products (k1. . .kn-1)*(h1s. . .hn-1s) and (k'1. . .k'n-1)*(h1s. . .hn-1s) of U(n-1) irreducible representations. Hence the proposed solution reflects in a direct way the operation of King's branching rule for the chain U(n)*U(n) contains/implies U(n), supplemented, whenever necessary, with King's modification rule.
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