Construction of hyperspherical functions symmetrized with respect to the orthogonal and the symmetric groups

We present a general recursive algorithm for the efficient construction ofN-body wave functions that belong to a given irreducible representation (irrep) of the orthogonal group and are at the same time characterized by a well-defined permutational symmetry. The main idea is to construct independently the hyperspherical functions with well defined orthogonal symmetry, and then reduce the irreps of the orthogonal group into the appropriate irreps of the symmetry group. The recursive algorithm in both groups is similar - we diagonalize the appropriate second order Casimir operator. The algorithm is applied to the hyperspherical functions, which are standard basis functions forN-body calculations. The evaluation of one and two-body matrix elements, in this basis, requires the use of the various hyperspherical coefficients, which are given in this paper. We have encoded this algorithm and found it very efficient for calculating symmetrized hyperspherical functions. We found that, in our method, the number of coefficients of fractional parentages involved is reduced drastically compared to previous methods which do not use the orthogonal group. Therefore we are able to construct the symmetrized basis functions forN-body systems that are beyond the reach of the other approaches.