Schubert polynomials and skew Schur functions

Abstract

There is an extensive literature about symmetric polynomials see e .g. Macdonald (1979), and many combinatorial algorithms are known for the computation with symmetric polynomials. On the other hand there is not much known for non symmetrical polynomials . Lascoux and Schiitzenberger (1982) defined Schubert polynomials, which generalize symmetric polynomialsand form a 7L-basis of the space of polynomials 7L[al, a2, . . .] . The combinatorics of these polynomials is very simliar to the combinatorics of symmetric polynomials. Moreover Schur functions, which are the most fundamental basis of the space of symmetric polynomials, are special Schubert polynomials . Skew Schur functions are not Schubert, but we show in this paper, that they are restrictions of Schubert polynomials. Lascoux and Schutzenberger gave an algorithm, called the transition formula in Lascoux and Schutzenberger (1985) to compute the decomposition of the symmetric part of the Schubert polynomial into a sum of Schur polynomials . They used this algorithm to compute the decomposition of the product of Schur functions as a sum of Schur functions, i .e. the computation of the Littlewood Richardson coefficents . We will use their algorithm to compute the decomposition of a skew Schur function into a sum of Schur functions . This method is totally different from the Littlewood Richardson rule, which provides the same decomposition . We use this method in the program SYMMETRICA, formerly called SYMCHAR, which is presented in the same volume, to compute the decomposition of skew Schur functions .