General Coupling Coefficients for the Group SU(3)

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Abstract
A Hilbert space method, previously applied to the group SU(2), is employed to examine the representations 𝒟λμ and the reduction of the direct-product representation of the group SU(3). The base vectors |λμ;α>, an orthogonal Hilbert space of homogeneous polynomials, are transformed to the base vectors |λμ;α>c, and are associated with the complex conjugate representation by an explicit R- conjugation operation. For the general direct-product representation 𝒟λ1μ1 ⊗ 𝒟λ2μ2, explicit expressions are derived for the vector coupling coefficients and the number of times the irreducible representation 𝒟λ3μ3 is contained in the direct product. Two methods of labeling the degenerate states are given, the reduction of the direct product is shown to be complete, and the symmetry relations of the 3(λμ) coefficients are discussed.