Split bases and multiplicity separations in symmetric group transformation coefficients

Abstract

We consider matrices transforming between the standard Young-Yamanouchi basis of the symmetric group and bases adapted to the product subgroups (the split basis). We derive closed formulae for transformation coefficients for b = 3, which includes the first cases when a choice of multiplicity separation is required. We discuss considerations which can be applied to obtain a simple form for the multiplicity separation. We show that the combinatorial and algebraic structure of the Littlewood-Richardson rule, also known as the Biedenharn-Louck pattern calculus, does not assist with finding a simple multiplicity separation.