  We focus on the tranformation matrices between the standard Young-Yamanouchi
basis of an irreducible representation for the symmetric group S_n and the
split basis adapted to the direct product subgroups S_{n_1} \times S_{n-n_1} .
We introduce the concept of subduction graph and we show that it conveniently
describes the combinatorial structure of the equation system arisen from the
linear equation method. Thus we can outline an improved algorithm to solve the
subduction problem in symmetric groups by a graph searching process. We
conclude observing that the general matrix form for multiplicity separations,
resulting from orthonormalization, can be expressed in terms of Sylvester
matrices relative to a suitable inner product in the multiplicity space.
