Coefficient of fractional parentage

In physics, coefficients of fractional parentage (cfp's) can be used to obtain anti-symmetric many-body states for like particles. In a jj-coupling scheme they are defined by the following relation.

$\Psi _{1\ldots N}(j^{N}\alpha JM)=\sum _{\alpha _{1}J_{1}}\langle j^{N-1}(\alpha _{1}J_{1});jJ\mid \}j^{N}\alpha J\rangle \left[\Psi _{1\ldots N-1}(j^{N-1}\alpha _{1}J_{1})\otimes \psi _{N}(j)\right]_{M}^{J}$

The state $\Psi _{1\ldots N}(j^{N}\alpha JM)$ is normalized and totally anti-symmetric with respect to permutations of all its N particles, while the state $\Psi _{1\ldots N-1}(j^{N-1}\alpha _{1}J_{1})$ is normalized and totally anti-symmetric with respect to all its N − 1 particles.