CHARACTER ANALYSIS OF U(N) AND SU(N)

A symmetric group analysis of the characters of U(N) and SU(N) representations yields formulas for (i) the multiplicities of weights in irreducible and tensor product representations, (ii) the coefficients occurring in the Clebsch-Gordan series decomposition of Kronecker products with an arbitrary number of factors, (iii) the content of irreducible and tensor product representations of U(∑i Ni) with respect to representations of its direct product subgroup, U(N1) ⊗ U(N2) ⊗ ⋯ ≡ ⊗i U(N i), and (iv) the content of irreducible representations of U(NM) with respect to irreducible representations of U(N) ⊗ U(M). In particular, we exhibit formulas for (i), (ii), and (iii) containing only irreducible characters and Frobenius compound characters of the symmetric group. Under the application of an operator of the subgroup, ⊗i U(N i) with ∑i Ni < N, a vector in a representation of U(N) transforms as a linear combination of vectors in irreducible representations of the subgroup. We give formulas for determining the vectors occurring in such a linear combination. They are derived in a similar fashion to the formulas for (i), (ii), and (iii). In terms of weight diagrams, the formulas give the number of times a weight diagram of the subgroup's algebra occurs in the hyperplane generated by the application of the algebra to the weight of the U(N) vector in question.