Direct Evaluation of Fractional Parentage Coefficients Using Young Operators. General Theory and〈4|2, 2〉Coefficients

Abstract

General expressions for $〈n|m, n-m〉$ fractional parentage coefficients are derived, for the special case where the state vectors are all different, by using normalized Young symmetry operators constructed out of permutation operators acting on the state vectors. The coefficients are given directly in terms of matrix elements of the permutations characterizing the cosets of the subgroup $S_m\times S_{n-m}$ of $S_n$ multiplied by the corresponding permutation operator. The permutation operators themselves are expressible in terms of Racah $W$ functions. Explicit expressions for the $〈4|2, 2〉$ coefficients are given also for cases where some of the state vectors are identical.