[Next] [Contents]

## The library `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.

[Next] [Contents]

MuPAD Combinat, an open source algebraic combinatorics package