A MAPLE program for calculations with Schur functions Original Research Article

We describe an interactive program, to be used in the MAPLE environment, for calculations involving Schur functions. The program performs Schur function operations of interest to applications in many-body physics: outer product, internal product, division and outer plethysm. Essential to the program is a new algorithm for the multiplication of monomial symmetric functions, which is used to implement the evaluation of outer product and outer plethysm. The same algorithm is extended to apply to the division of monomial symmetric functions, with a novel use of partitions containing negative integers, useful for implementing the evaluation of division. In particular, the evaluation of outer plethysms involving sums, differences, and products of Schur functions with this program is as simple as those involving only single Schur functions. The program can also be used to expand a Schur function in terms of monomial symmetric functions or expand a monomial symmetric function in terms of Schur functions, i.e. to calculate the Kostka and inverse Kostka numbers. The calculation of Kostka numbers makes substantial use of Gelʹfand patterns. The number of variables in which Schur functions or monomial symmetric functions are expressed can be specified in advance by the user, which results in significant reduction of calculation time.