Recursive construction of irreducible tensor spaces of unitary groups and fractional parentage expansion of single-configuration wave function

It is shown that irreducible tensor spaces of the unitary group $U_r$ with special bases fixed by a definite choice of the embedded chain of unitary subgroups $U_{r_1}+\dots+U_{r_k}$ $(r_1+\dots+r_k=r)$, each of which, in its turn, is reduced on an arbitrary group $G$, can be constructed recursively by means of the Clebsch–Gordan coefficients of the group $U_r$. These coefficients are factorized and the relationship between the recursive construction of the irreducible tensor spaces and the fractional parentage decomposition of a single-configuration wave function is established.