SymmetricPolynomials -- the algebra of symmetric polynomials

Description

This package uses an explicit description of the Groebner basis of the ideal of obvious relations in this algebra based on:

Grayson, Stillmann - Computations in the intersection theory of flag varieties, preprint, 2009

Sturmfels - Algorithms in Invariant Theory, Springer Verlag, Vienna, 1993

Use elementarySymmetric to express a symmetric polynomial in terms of elementary symmetric functions, and use elementarySymmetric(PolynomialRing) to construct the corresponding map.

i1 : n=5;

i2 : R=QQ[x_1..x_n];

i3 : f=(product gens R)*(sum gens R);

i4 : elementarySymmetric f

o4 = e e
      1 5

o4 : QQ[x ..x , e ..e ]
         1   5   1   5

i5 : elementarySymmetric R

o5 = map (QQ[x ..x , e ..e ], R, {x , x , x , x , x })
              1   5   1   5        1   2   3   4   5

o5 : RingMap QQ[x ..x , e ..e ] <-- R
                 1   5   1   5

Author

Alexandra Seceleanu

Version

This documentation describes version 1.0 of SymmetricPolynomials, released May 20 2009.

Citation

If you have used this package in your research, please cite it as follows:

@misc{SymmetricPolynomialsSource,
  title = {{SymmetricPolynomials: A \emph{Macaulay2} package. Version~1.0}},
  author = {Alexandra Seceleanu},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- buildSymmetricGB -- Groebner basis of elementary symmetric polynomials algebra
- elementarySymmetric -- expression in terms of elementary symmetric polynomials
Methods
- buildSymmetricGB(PolynomialRing) -- Groebner basis of elementary symmetric polynomials algebra
- elementarySymmetric(PolynomialRing) -- elementary symmetric polynomials algebra
- elementarySymmetric(RingElement) -- expression in terms of elementary symmetric polynomials

For the programmer

The object SymmetricPolynomials is a package, defined in SymmetricPolynomials.m2.

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/SymmetricPolynomials.m2:100:0.