A complementary group technique for the resolution of the outer multiplicity problem of SU(n): (II) A recoupling approach to the solution of SU(3)\supset U(2) reduced Wigner coefficients

Abstract

A general procedure for the derivation of SU(3)\supset U(2) reduced Wigner coefficients for the coupling (λ_{1}μ_{1})\times (λ_{2}μ_{2})\downarrow (λμ)^η, where ηis the outer multiplicity label needed in the decomposition, is proposed based on a recoupling approach according to the complementary group technique given in (I). It is proved that the non-multiplicity-free reduced Wigner coefficients of SU(n) are not unique with respect to canonical outer multiplicity labels, and can be transformed from one set of outer multiplicity labels to another. The transformation matrices are elements of SO(m), where m is the number of occurrence of the corresponding irrep (λμ) in the decomposition (λ_{1}μ_{1})\times (λ_{2}μ_{2})\downarrow (λμ). Thus, a kind of the reduced Wigner coefficients with multiplicity is obtained after a special SO(m) transformation. New features of this kind of reduced Wigner coefficients and the differences from the reduced Wigner coefficients with other choice of the multiplicity label given previously are discussed. The method can also be applied to the derivation of general SU(n) Wigner or reduced Wigner coefficients with multiplicity. Algebraic expression of another kind of reduced Wigner coefficients, the so-called reduced auxiliary Wigner coefficients for SU(3)\supset U(2), are also obtained.