Nth rank tensor representations of U(n) symmetry adapted to subgroups of the symmetric group S N

Representations of the symmetric group S_N symmetry adapted to subgroup sequences $\begin{array}{c}{\mathrm{\otimes S}}_{N_j}\\ \mathrm{\nearrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[SE\; pointing\; arrow]}\\ {\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}S}}_N{{\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\otimes S}}_{i^N}}_j\\ \mathrm{[SE\; pointing\; arrow]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\nearrow }\\ {\mathrm{\otimes S}}_{i^N}\end{array}$ are considered using double‐coset decomposition. The matrix elements of the double‐coset representatives are given and their group theoretical properties are discussed. The matrix elements are identified with the recoupling transformations of the unitary group by considering the tensor representations of the latter. The orthogonality and completeness relations of the symmetric group expressed in terms of double‐coset representative matrix elements are used to establish general relations that must be satisfied by the coupling coefficients of the unitary group.