This is SymmetricPolynomials.info, produced by Macaulay2, version 1.25.11

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

Author
======

  * Alexandra Seceleanu (see https://aseceleanu.github.io/ )

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 (Macaulay2Doc),
defined in SymmetricPolynomials.m2.

-------------------------------------------------------------------------------

The source of this document is in SymmetricPolynomials.m2:91:0.

buildSymmetricGB -- Groebner basis of elementary symmetric polynomials algebra
******************************************************************************

  * Usage:
        buildSymmetricGB R
  * Inputs:
      * R, aRing (Macaulay2Doc),
  * Outputs:
      * the Groebner basis of the elementary symmetric algebra

Description
===========

+-----------------------------------------------------------------------------+
|i1 : n=5;                                                                    |
+-----------------------------------------------------------------------------+
|i2 : R=QQ[x_1..x_n];                                                         |
+-----------------------------------------------------------------------------+
|i3 : buildSymmetricGB R                                                      |
|                                                                             |
|         5    4      3      2                 4    3      3      2 2    2    |
|o3 = {- x  + x e  - x e  + x e  - x e  + e , x  + x x  - x e  + x x  - x x e |
|         5    5 1    5 2    5 3    5 4    5   4    4 5    4 1    4 5    4 5 1|
|     ------------------------------------------------------------------------|
|        2        3      2                      4    3      2                 |
|     + x e  + x x  - x x e  + x x e  - x e  + x  - x e  + x e  - x e  + e , -|
|        4 2    4 5    4 5 1    4 5 2    4 3    5    5 1    5 2    5 3    4   |
|     ------------------------------------------------------------------------|
|      3    2      2      2        2                        2                 |
|     x  - x x  - x x  + x e  - x x  - x x x  + x x e  - x x  + x x e  - x e  |
|      3    3 4    3 5    3 1    3 4    3 4 5    3 4 1    3 5    3 5 1    3 2 |
|     ------------------------------------------------------------------------|
|        3    2      2        2                    3    2                 2   |
|     - x  - x x  + x e  - x x  + x x e  - x e  - x  + x e  - x e  + e , x  + |
|        4    4 5    4 1    4 5    4 5 1    4 2    5    5 1    5 2    3   2   |
|     ------------------------------------------------------------------------|
|                                  2                         2                |
|     x x  + x x  + x x  - x e  + x  + x x  + x x  - x e  + x  + x x  - x e  +|
|      2 3    2 4    2 5    2 1    3    3 4    3 5    3 1    4    4 5    4 1  |
|     ------------------------------------------------------------------------|
|      2                                                                      |
|     x  - x e  + e , - x  - x  - x  - x  - x  + e }                          |
|      5    5 1    2     1    2    3    4    5    1                           |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+

This function should work up to a size of 15 variables in the base ring

This function is part of the package SymmetricPolynomials.

Ways to use buildSymmetricGB:
=============================

  * buildSymmetricGB(PolynomialRing)
    -- Groebner basis of elementary
    symmetric polynomials algebra

For the programmer
==================

The object buildSymmetricGB, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in SymmetricPolynomials.m2:155:0.

buildSymmetricGB(PolynomialRing) -- Groebner basis of elementary symmetric polynomials algebra
**********************************************************************************************

  * Function: buildSymmetricGB,
  * Usage:
        buildSymmetricGB R
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), a
        PolynomialRing (Macaulay2Doc),
  * Outputs:
      * the Groebner basis of the elementary symmetric algebra

Description
===========

+-----------------------------------------------------------------------------+
|i1 : n=5;                                                                    |
+-----------------------------------------------------------------------------+
|i2 : R=QQ[x_1..x_n];                                                         |
+-----------------------------------------------------------------------------+
|i3 : buildSymmetricGB R                                                      |
|                                                                             |
|         5    4      3      2                 4    3      3      2 2    2    |
|o3 = {- x  + x e  - x e  + x e  - x e  + e , x  + x x  - x e  + x x  - x x e |
|         5    5 1    5 2    5 3    5 4    5   4    4 5    4 1    4 5    4 5 1|
|     ------------------------------------------------------------------------|
|        2        3      2                      4    3      2                 |
|     + x e  + x x  - x x e  + x x e  - x e  + x  - x e  + x e  - x e  + e , -|
|        4 2    4 5    4 5 1    4 5 2    4 3    5    5 1    5 2    5 3    4   |
|     ------------------------------------------------------------------------|
|      3    2      2      2        2                        2                 |
|     x  - x x  - x x  + x e  - x x  - x x x  + x x e  - x x  + x x e  - x e  |
|      3    3 4    3 5    3 1    3 4    3 4 5    3 4 1    3 5    3 5 1    3 2 |
|     ------------------------------------------------------------------------|
|        3    2      2        2                    3    2                 2   |
|     - x  - x x  + x e  - x x  + x x e  - x e  - x  + x e  - x e  + e , x  + |
|        4    4 5    4 1    4 5    4 5 1    4 2    5    5 1    5 2    3   2   |
|     ------------------------------------------------------------------------|
|                                  2                         2                |
|     x x  + x x  + x x  - x e  + x  + x x  + x x  - x e  + x  + x x  - x e  +|
|      2 3    2 4    2 5    2 1    3    3 4    3 5    3 1    4    4 5    4 1  |
|     ------------------------------------------------------------------------|
|      2                                                                      |
|     x  - x e  + e , - x  - x  - x  - x  - x  + e }                          |
|      5    5 1    2     1    2    3    4    5    1                           |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+

This function should work up to a size of 15 variables in the base ring

This function is part of the package SymmetricPolynomials.

Ways to use this method:
========================

  * buildSymmetricGB(PolynomialRing)
    -- Groebner basis of elementary
    symmetric polynomials algebra
-------------------------------------------------------------------------------

The source of this document is in SymmetricPolynomials.m2:170:0.

elementarySymmetric -- expression in terms of elementary symmetric polynomials
******************************************************************************

  * Usage:
        elementarySymmetric f
  * Inputs:
      * f, a symmetric
      * RingElement (Macaulay2Doc)
  * Outputs:
      * the expression of f in terms of the elementary symmetric functions e_i

Description
===========

+-------------------------------------+
|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               |
+-------------------------------------+

This function should work up to a size of 15 variables in the base ring.

This function is part of the package SymmetricPolynomials.

Caveat
======

if the input is not symmetric the function will announce this

Ways to use elementarySymmetric:
================================

  * elementarySymmetric(PolynomialRing)
    -- elementary symmetric
    polynomials algebra
  * elementarySymmetric(RingElement)
    -- expression in terms of elementary
    symmetric polynomials

For the programmer
==================

The object elementarySymmetric, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in SymmetricPolynomials.m2:108:0.

elementarySymmetric(PolynomialRing) -- elementary symmetric polynomials algebra
*******************************************************************************

  * Function: elementarySymmetric,
  * Usage:
        elementarySymmetric R
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), a
        PolynomialRing (Macaulay2Doc),
  * Outputs:
      * a map from R adjoin the elementary symmetric functions e_i to R

Description
===========

+------------------------------------------------------+
|i1 : n=5;                                             |
+------------------------------------------------------+
|i2 : R=QQ[x_1..x_n];                                  |
+------------------------------------------------------+
|i3 : elementarySymmetric R                            |
|                                                      |
|o3 = map (QQ[x ..x , e ..e ], R, {x , x , x , x , x })|
|              1   5   1   5        1   2   3   4   5  |
|                                                      |
|o3 : RingMap QQ[x ..x , e ..e ] <-- R                 |
|                 1   5   1   5                        |
+------------------------------------------------------+

This function should work up to a size of 15 variables in the base ring

This function is part of the package SymmetricPolynomials.

Ways to use this method:
========================

  * elementarySymmetric(PolynomialRing)
    -- elementary symmetric
    polynomials algebra
-------------------------------------------------------------------------------

The source of this document is in SymmetricPolynomials.m2:140:0.

elementarySymmetric(RingElement) -- expression in terms of elementary symmetric polynomials
*******************************************************************************************

  * Function: elementarySymmetric,
  * Usage:
        elementarySymmetric f
  * Inputs:
      * f, a ring element (Macaulay2Doc), a symmetric
        RingElement (Macaulay2Doc),
  * Outputs:
      * the expression of f in terms of the elementary symmetric functions e_i

Description
===========

+-------------------------------------+
|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               |
+-------------------------------------+

This function should work up to a size of 15 variables in the base ring

This function is part of the package SymmetricPolynomials.

Caveat
======

if the input is not symmetric the function will announce this

Ways to use this method:
========================

  * elementarySymmetric(RingElement)
    -- expression in terms of elementary
    symmetric polynomials
-------------------------------------------------------------------------------

The source of this document is in SymmetricPolynomials.m2:125:0.