The decomposition of the tensor product of representations of the symmetric group

Abstract

The decomposition of the tensor product of two irreducible representations of the symmetric group is studied, providing a foundation for the calculation of the Clebsch–Gordon coefficients. Realizations on spaces of polynomials are emphasized. This leads to tensor coupling coefficients. An iterative formula for the calculation of the tensor coupling coefficients is derived, and symmetry constructions are discussed and shown to lead to a reduction in the calculation needed. A well‐defined procedure for the construction of the Clebsch–Gordan coefficients from the tensor coupling coefficients, which works for any multiplicity, is obtained. It may be used for any finite group and, with some modifications, for any compact group.