Transformation coefficients between { \it standard} bases for irreducible representations of the symmetric group $S_n$ and { \it split} bases adapted to the $S_{ n_1} \times S_{ n_2} \subset S_n$ subgroup ($n_1 +n_2 = n$) are considered. We first provide a \emph{ selection rule} and an \emph{ identity rule} for the subduction coefficients which allow to decrease the number of unknowns and equations arising from the linear method by Pan and Chen. Then, using the { \it reduced subduction graph} approach, we may look at higher multiplicity instances. As a significant example, an orthonormalized solution for the first multiplicity-three case, which occurs in the decomposition of the irreducible representation $[4,3,2,1]$ of $S_{ 10}$ into $[3,2,1] \otimes [3,1]$ of $S_6 \times S_4$, is presented and discussed.
Categories: math-ph hep-th math.MP