Summation relation for U(N) Racah coefficients

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Abstract
A summation relation is given for U(N) Racah coefficients which has the form of an orthogonality relation, or a composition of recoupling transformations, except that the summation over column indices (for fixed row indices) is over multiplicity labels only. In the recoupling matrix for [f1] × [f2] × [f3] → [f], U(N) irreducible representations [f2] and [f3] are limited to be elementary, [11…10…0]≡[1k], or totally symmetric [k], or of the form [kN−1]. Results are tabulated as functions of the axial distances in [f] for [f2]=[1N−1], [1N−2], or [2N−1]; [f3]=[1], [12], or [2]; all cases which arise in the evaluation of squares of matrix elements of one‐ and two‐body operators averaged over irreducible representations of U(N)