Plethysms of Schur functions and the shell model

Abstract

We present a method for evaluating plethysms of Schur functions that is conceptually simpler than existing methods. Moreover the algorithm can be easily implemented with an algebraic computer language. Plethysms of sums, differences and products of S-functions are dealt with in exactly the same manner as plethysms of simple S-functions. Sums and differences of S-functions are of importance for the description of multi-shell configurations in the shell model. The number of variables in which the S-functions are expressed can be specified in advance, significantly simplifying the calculations in typical applications to many-body problems. The method relies on an algorithm that we have developed for the product of monomial symmetric functions. We present a new way of calculating the Kostka numbers (using Gel'fand patterns) and give, as well, a new formula for the Littlewood-Richardson coefficients.