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 xi 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@kfunigraz.ac.at,
last changed: November 19, 2001

 Multivariate polynomials