General Approach to Fractional Parentage Coefficients

The purpose of this paper is to achieve a clearer understanding of the problems involved in the determination of a closed formula for fractional parentage coefficients (fpc). The connection between the fpc and one‐block Wigner coefficients of a unitary group of dimension equal to that of the number of states is explicitly derived. Furthermore, these Wigner coefficients are decomposed into ones characterized by a canonical chain of subgroups (for which an explicit formula is given) and transformation brackets from the canonical to the physical chain. It is in the explicit and systematic determination of the states in the latter chain where the main difficulty appears. We fully analyze the case of the p shell to show that a complete nonorthonormal set of states in the physical chain $\mathcal{U}\text{(3)⊃R(3)}$ can be derived easily using Littlewood's procedure for the reduction of irreducible representations (IR) of SU(3) with respect to the subgroup R(3). This procedure gives a deeper understanding of the free exponent appearing in the polynomials in the creation operators defining the states in the $\mathcal{U}\text{(3)⊃R(3)}$ chain. As Littlewood's procedure applies to the $\mathcal{U}\mathrm{(n)\supset R(n)}$ chain, and probably can be generalized to other noncanonical chains of groups, it opens the possibility of obtaining general closed formulas for the fpc in a nonorthonormal basis.