SYMMETRICA-MANUAL -- Multivariate polynomials

Multivariate 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):

Example:
scan(PERMUTATION,a);
m_perm_schubert_monom_summe(a,b);
m_perm_schubert_qpolynom(a,c);
m_perm_schubert_dimension(a,d);
In case you enter the permutation [2,4,3,5,1], then the output looks as follows:
[2,4,3,5,1]

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

1 [6] 1 [7]

2

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

x₁x₂²x₃x₄+x₁²x₂x₃x₄,

that we obtain (by replacing x_i by qⁱ) the specialization

q⁶+q⁷,

and that the sum of all the coefficients is

2.

In order to write a given multivariate polynomial as a linear combination of Schubert polynomials you can use

t_POLYNOM_SCHUBERT

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:

t_SCHUBERT_POLYNOM

The corresponding scan(SCHUBERT,a) works in an analogous way as scan(SCHUR,a), instead of partition you use permutations in list notation.

harald.fripertinger "at" uni-graz.at, May 26, 2011

Multivariate polynomials