muEC::SP – mu-EC library for Schubert polynomials (package Combinat, deprecated)

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 yi'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 ementary symmetric functions on a flag of alphabets.

muEC::SP::NewtonInterp – Newton interpolation formula

muEC::SP::Sp2TableVar – variable set for the SP package

muEC::SP::SpScalarPol – scalar product on polynomials

muEC::SP::SpSpecialize – specializes a set of variables

muEC::SP::TablePe – table of the Pe-basis

muEC::SP::TableX – table of all Schubert polynomials

muEC::SP::TableXX – table of all double Schubert polynomials

muEC::SP::TableY – table of all Schubert polynomials

muEC::SP::TableYY – table of all double Schubert polynomials

muEC::SP::ToPe – converts any expression to the Pe-basis

muEC::SP::ToX – converts any expression to the X Schubert basis

muEC::SP::ToXX – converts any expression to the XX Schubert basis

muEC::SP::ToY – converts any expression to the Y Schubert basis

muEC::SP::ToYY – converts any expression to the YY Schubert basis

muEC::SP::Tox – expresses any expression in the basis of monomials

muEC::SP::X2Dim – specialization in X Schubert basis

muEC::SP::x2X – from the basis of monomials to X Schubert basis

muEC::SP::x2XX – from the basis of monomials to XX Schubert basis