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Hyperspherical Function and SP(2,R) Group

In the present paper a method for constructing the hyperspherical function for the internal motion of many-body system in the scheme Sp(2m,R) ⊃ Sp(2,R) × O(m) based on the complementarity of the symplectic SP(2,R) and the orthogonal O(m) groups is developed. A generate formula for the fractional-parentage coefficients of the hyperspherical function is obtained and it is shown that the formula can be used to to calculate the spectroscopic factor for the arbitrary m-cluster, etc..