Representations of the Symmetric Group and the Fractional Parentage Coefficients

The fractional parentage coefficients are treated by using the Young operator of the symmetric group and general expressions for the coefficients are obtained. From these expressions one can evaluate the fractional parentage coefficients which separate any numbers of particles from the others for pure or mixed configurations. In order to calculate the fractional parentage coefficients which split off more than one particles, non-standard representations of the symmetric group are made use of. The representations are the outer products of two representations of the symmetric group and the transformation coefficients with the standard representations are calculated.