Recoupling coefficients of the symmetric group for shell and cluster model configurations

Recoupling coefficients of the symmetric group are defined from the reduction of irreducible representations of this group to direct product representations of particular subgroups. These recoupling coefficients are analysed and shown to be identical to the invariants of unitary groups arising from the reduction of Kronecker product representations. The recoupling coefficients have applications in configurations of several shells or clusters.