Branching Laws, Inner Multiplicities, and Decomposition of Classical Groups

A method has been found whereby the inner multiplicity of all classical groups [all irreducible representations of SU(n) and SO(2k + 1) and some simple irreducible representations of SO(2k) and Sp(2k)] can be obtained easily from the branching laws of Weyl (unitary), Boerner (orthogonal), and Hegerfeldt (symplectic). Once the inner multiplicity is known, the same formula can be used again to obtain the decomposition of a classical group into its subgroups without any restriction. Finally, since the inner multiplicity is connected to the outer multiplicity through the Racah-Speiser lemma, this method enables us to obtain the Clebsch-Gordan series for the direct product of all classical groups.