Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients

Abstract

A formalism of decomposing tensor product of irreducible representations of compact semisimple Lie groups is presented, using holomorphic induction techniques. The case of SU(n) is examined in detail. Irreducible representation spaces are realized as polynomial functions over GL(n,ℂ) group variables and it is shown how to generate invariant spaces labelled by double cosets, with one double coset subspace isomorphic to the original tensor product space. Global and noninductive procedure for constructing orthogonal polynomial bases is presented and is compared with the Gelfand-Žetlin procedure. Clebsch-Gordan (Wigner) coefficients are computed and are used to provide a resolution of the multiplicity problem occurring in the tensor product decompositions of U(n) groups.