Complementary group resolution of the $SU(n)$ outer multiplicity problem. II. Recoupling approach for SU(3)⊃U(2) reduced Wigner coefficients

A general procedure for the derivation of SU(3)⊃U(2) reduced Wigner coefficients (RWCs) for the coupling $(λ1μ1)×(λ2μ2)↓(λμ)η,$ where η is the outer multiplicity label required in the decomposition, is proposed based on a recoupling approach that follows the complementary group technique for a resolution of the outer multiplicity of $SU(n)$ introduced in Part (I) of this series. RWCs of $SU(n)$ are not unique under a canonical resolution of the outer multiplicity; the transformation from one set to another are elements of SO(m), where m is the number of occurrences of the (λμ) irrep in the decomposition $(λ1μ1)×(λ2μ2)↓(λμ).$ A special resolution of the multiplicity is identified that leads to a recursive procedure for the determination of RWCs. New features of these special RWCs and differences from those obtained with other choices are discussed. The method can be applied to the derivation of general $SU(n)$ Wigner or RWCs. Algebraic expressions for another kind of RWCs, the so-called reduced auxiliary Wigner coefficients for SU(3)⊃U(2), are also obtained.