Finite group actionsMixed examplesSymmetric polynomialsMultivariate polynomials

Multivariate polynomials

As it was mentioned already, the routine for the evaluation of Littlewood-Richardson coefficients uses the calculus of Schubert polynomials as it was suggested by Lascoux and Schützenberger. These polynomials are associated with the permutations of N, they form a basis of Z[x1,x2,...], and you can decompose a multivariate polynomial in this basis! Therefore you can switch from the monomial basis to the basis of Schubert polynomials. Here are the main lines of a corresponding program which first evaluates the Schubert polynomial associated with the scanned permutation, then it gives the polynomial in a single indeterminate q which arises by replacing the i-th indeterminate of the Schubert polynomial by qi, while the final line replaces each monomial summand by 1 so that the result is the sum of the coefficients of the monomials (a complete program is stored in ex11.c):
In case you enter the permutation [2,4,3,5,1], then the output looks as follows:


1 [1,2,1,1,0] 1 [2,1,1,1,0]

1 [6] 1 [7]


This means that the Schubert polynomial corresponding to the above permutation is

that we obtain (by replacing xi by qi) the specialization
and that the sum of all the coefficients is
In order to write a given multivariate polynomial as a linear combination of Schubert polynomials you can use
Here the output is a linear combination of permutations (=Schubert polynomials). Conversely you can input a linear combination of permutations and obtain the corresponding polynomials which is the linear combination of the Schubert polynomials associated with the permutations:
The corresponding scan(SCHUBERT,a) works in an analogous way as scan(SCHUR,a), instead of partition you use permutations in list notation.,
last changed: November 19, 2001

Finite group actionsMixed examplesSymmetric polynomialsMultivariate polynomials