muEC::SP
The muEC::SP
library provides functions to work on simple and
double Schubert polynomials. Schubert polynomials form a linear
basis of the ring of multivariate polynomials.
All defined bases are stored in the global variable
muEC::SP::SPBases
that is by default [x, y, Pe, X, Y, XX,
YY]
. Note that the y
i's are seen as special variables
since they are the second alphabet for double Schubert polynomials.
Schubert polynomials may be indexed by permutations or integer
vectors. In the first case, they are denoted by the expression
X[
permutation]
and in the second one, by
Y[
code]
where code is a sequence of nonnegative integers
(interpreted as the code of a permutation).
In the case of double Schubert polynomials we denote by
XX[
permutation]
the double Schubert polynomial indexed
by a permutation and by YY[
code]
the corresponding
double Schubert polynomial indexed by a vector of non-negative
integers (interpreted as the code of a permutation).
The Pe
-basis is the dual basis of the basis of increasing
monomials (less than the `staircase' monomial), according to the
scalar product SP::SpScalarPol
. It is the basis of products
of elementary symmetric functions on a flag of alphabets.
MuPAD Combinat, an open source algebraic combinatorics package