This is SLnEquivariantMatrices.info, produced by Macaulay2, version 1.22

SLnEquivariantMatrices -- Ancillary file to the paper "A construction of equivariant bundles on the space of symmetric forms"
*****************************************************************************************************************************

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

In the paper "A construction of equivariant bundles on the space of symmetric
forms" (https://arxiv.org), the authors construct stable vector bundles on the
space $\PP(S^d\CC^{n+1})$ of symmetric forms of degree $d$ in $n + 1$ variables
which are equivariant for the action of $SL_{n+1}(\CC)$ ,and admit an
equivariant free resolution of length 2.

Take two integers $d \ge 1$ and $m \ge 2$ and a vector spave $V = \CC^{n+1}$.
For $n=2$, we have

$S^dV \otimes S^{(m-1)d}V = S^{md}V \oplus S^{md-2}V \oplus S^{md-4}V \oplus
\dots$,

while for $n > 1$,

$S^dV \otimes S^{(m-1)d}V = S^{md}V \oplus V_{(md-2)\lambda_1+\lambda_2} \oplus
V_{(md-4)\lambda_1+2\lambda_2} \oplus \dots$,

where $\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of
the Lie group $SL_{n+1}(\bf C)$ and $V_{i\lambda_1+j\lambda_2}$ is the
irreducible representation of highest weight $i\lambda_1+j\lambda_2$.

The projection of the tensor product onto the second summand induces a
$SL_{2}(\CC)$-equivariant morphism

$\Phi: S^{md-2}V \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V \otimes
O_{\PP(S^dV)}(1)$

or a $SL_{n+1}(\CC)$-equivariant morphism

$\Phi: V_{(md-2)\lambda_1 + \lambda_2} \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V
\otimes O_{\PP(S^dV)}(1)$

with constant co-rank 1, and thus gives an exact sequence of vector bundles on
$\PP(S^dV)$:

$0 \to W_{2,d,m} \to S^{md-2}V \otimes \mathcal{O}_{\PP(S^dV)} \to S^{(m-1)d}V
\otimes \mathcal{O}_{\PP(S^dV)}(1) \to \mathcal{O}_{\PP(S^dV)}(m) \to 0$,

$0 \to W_{n,d,m} \to V_{(md-2)\lambda_1 + \lambda_2} \otimes
\mathcal{O}_{\PP(S^dV)} \to S^{(m-1)d}V \otimes \mathcal{O}_{\PP(S^dV)}(1) \to
\mathcal{O}_{\PP(S^dV)}(m) \to 0$.

The package allows to compute

(1) the decomposition into irreducible $SL_{n+1}(\bf C)$-representations of the
tensor product of two symmetric powers $S^a\CC^{n+1}$ and $S^b\CC^{n+1}$;

(2) the matrix representing the morphism $\Phi$;

(3) the vector bundle $W_{n,d,m}$.

Authors
=======

  * Ada Boralevi (see http://www.adaboralevi.com ) <ada.boralevi@polito.it>
  * Daniele Faenzi (see http://dfaenzi.perso.math.cnrs.fr )
    <daniele.faenzi@u-bourgogne.fr>
  * Paolo Lella (see http://www.paololella.it ) <paolo.lella@polimi.it>

Version
=======

This documentation describes version 1.0 of SLnEquivariantMatrices.

Source code
===========

The source code from which this documentation is derived is in the file
SLnEquivariantMatrices.m2 (see common/share/Macaulay2/SLnEquivariantMatrices.m2
).

Exports
=======

  * Functions and commands
      * sl2EquivariantConstantRankMatrix
        -- computes a SL(2)-equivariant
        constant rank matrix
      * sl2EquivariantVectorBundle, --
        computes a SL(2)-equivariant vector bundle over some projective space
      * slEquivariantConstantRankMatrix,
        -- computes a SL-equivariant constant rank matrix
      * slEquivariantVectorBundle -- computes
        a SL-equivariant vector bundle over some projective space
      * slIrreducibleRepresentationsTensorProduct
        -- computes the irreducible
        SL-subrepresentations of the tensor product of two symmetric products
  * Methods
      * "sl2EquivariantConstantRankMatrix(PolynomialRing,ZZ)" -- see
        sl2EquivariantConstantRankMatrix, --
        computes a SL(2)-equivariant constant rank matrix
      * "sl2EquivariantConstantRankMatrix(ZZ,ZZ)" -- see
        sl2EquivariantConstantRankMatrix, --
        computes a SL(2)-equivariant constant rank matrix
      * "sl2EquivariantVectorBundle(PolynomialRing,ZZ)" -- see
        sl2EquivariantVectorBundle, -- computes a
        SL(2)-equivariant vector bundle over some projective space
      * "sl2EquivariantVectorBundle(ZZ,ZZ)" -- see
        sl2EquivariantVectorBundle, -- computes a
        SL(2)-equivariant vector bundle over some projective space
      * "slEquivariantConstantRankMatrix(PolynomialRing,ZZ,ZZ)" -- see
        slEquivariantConstantRankMatrix, --
        computes a SL-equivariant constant rank matrix
      * "slEquivariantConstantRankMatrix(PolynomialRing,ZZ,ZZ,PolynomialRing)"
        -- see slEquivariantConstantRankMatrix,
        -- computes a SL-equivariant constant
        rank matrix
      * "slEquivariantConstantRankMatrix(ZZ,ZZ,ZZ)" -- see
        slEquivariantConstantRankMatrix, --
        computes a SL-equivariant constant rank matrix
      * "slEquivariantConstantRankMatrix(ZZ,ZZ,ZZ,PolynomialRing)" -- see
        slEquivariantConstantRankMatrix, --
        computes a SL-equivariant constant rank matrix
      * "slEquivariantVectorBundle(PolynomialRing,ZZ,ZZ)" -- see
        slEquivariantVectorBundle, -- computes a
        SL-equivariant vector bundle over some projective space
      * "slEquivariantVectorBundle(PolynomialRing,ZZ,ZZ,PolynomialRing)" -- see
        slEquivariantVectorBundle, -- computes
        a SL-equivariant vector bundle over some projective space
      * "slEquivariantVectorBundle(ZZ,ZZ,ZZ)" -- see
        slEquivariantVectorBundle, -- computes a
        SL-equivariant vector bundle over some projective space
      * "slEquivariantVectorBundle(ZZ,ZZ,ZZ,PolynomialRing)" -- see
        slEquivariantVectorBundle, -- computes a
        SL-equivariant vector bundle over some projective space
      * "slIrreducibleRepresentationsTensorProduct(PolynomialRing,ZZ,ZZ)" --
        see slIrreducibleRepresentationsTensorProduct,
        -- computes the irreducible
        SL-subrepresentations of the tensor product of two symmetric products
      * "slIrreducibleRepresentationsTensorProduct(ZZ,ZZ,ZZ)" -- see
        slIrreducibleRepresentationsTensorProduct,
        -- computes the irreducible
        SL-subrepresentations of the tensor product of two symmetric products

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

The object SLnEquivariantMatrices, is a package (Macaulay2Doc).

sl2EquivariantConstantRankMatrix -- computes a SL(2)-equivariant constant rank matrix
*************************************************************************************

Synopsis
========

  * Usage:
        M = sl2EquivariantConstantRankMatrix(d,m)
        M = sl2EquivariantConstantRankMatrix(d,m,CoefficientRing=>C)
        M = sl2EquivariantConstantRankMatrix(R,m)
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), with at
        least two variables
      * d, an integer (Macaulay2Doc), positive
      * m, an integer (Macaulay2Doc), at least 2
      * C, a ring (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing,
        => ..., default value QQ, name for optional argument
  * Outputs:
      * M, a matrix (Macaulay2Doc),

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

This function returns a constant rank matrix of linear forms. The matrix
describes the morphism

$\Phi: S^{md-2}V \otimes O_{\PP^d} \to S^{(m-1)d}V \otimes O_{\PP^d)}(1)$

given by the projection

$S^dV \otimes S^{(m-1)d}V \to S^{md-2}V$

of the irreducible $SL(2)$-subrepresentation of highest weight $md-2$, where
$\PP^d = \PP(S^dV)$ as $V=<v_0,v_1>$. In the paper A construction of
equivariant bundles on the space of symmetric forms, the entries of the matrix
$\Phi$ are explicitly described.

+-----------------------------------------------------------------------------+
|i1 : d = 4, m = 3                                                            |
|                                                                             |
|o1 = (4, 3)                                                                  |
|                                                                             |
|o1 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i2 : M = sl2EquivariantConstantRankMatrix(d,m)                               |
|                                                                             |
|o2 = {1} | -x_1 -3x_2 -x_3  -x_4   0     0     0      0      0     0     0   |
|     {1} | x_0  -4x_1 -6x_2 -20x_3 -x_4  0     0      0      0     0     0   |
|     {1} | 0    7x_0  0     -42x_2 -8x_3 -x_4  0      0      0     0     0   |
|     {1} | 0    0     7x_0  28x_1  -6x_2 -4x_3 -5x_4  0      0     0     0   |
|     {1} | 0    0     0     35x_0  10x_1 0     -10x_3 -35x_4 0     0     0   |
|     {1} | 0    0     0     0      5x_0  4x_1  6x_2   -28x_3 -7x_4 0     0   |
|     {1} | 0    0     0     0      0     x_0   8x_1   42x_2  0     -7x_4 0   |
|     {1} | 0    0     0     0      0     0     x_0    20x_1  6x_2  4x_3  -x_4|
|     {1} | 0    0     0     0      0     0     0      x_0    x_1   3x_2  x_3 |
|     ------------------------------------------------------------------------|
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|                                                                             |
|                        9                  11                                |
|o2 : Matrix (QQ[x ..x ])  <--- (QQ[x ..x ])                                  |
|                 0   4              0   4                                    |
+-----------------------------------------------------------------------------+

By default, see sl2EquivariantConstantRankMatrix,
defines the matrix over a polynomial ring
with rational coefficients. The optional argument CoefficientRing (Macaulay2Doc),
allows one to change the coefficient ring.

+-----------------------------------------------------------------------------+
|i3 : d = 4, m = 3                                                            |
|                                                                             |
|o3 = (4, 3)                                                                  |
|                                                                             |
|o3 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i4 : M = sl2EquivariantConstantRankMatrix(d,m,CoefficientRing=>ZZ/10007)     |
|                                                                             |
|o4 = {1} | -x_1 -3x_2 -x_3  -x_4   0     0     0      0      0     0     0   |
|     {1} | x_0  -4x_1 -6x_2 -20x_3 -x_4  0     0      0      0     0     0   |
|     {1} | 0    7x_0  0     -42x_2 -8x_3 -x_4  0      0      0     0     0   |
|     {1} | 0    0     7x_0  28x_1  -6x_2 -4x_3 -5x_4  0      0     0     0   |
|     {1} | 0    0     0     35x_0  10x_1 0     -10x_3 -35x_4 0     0     0   |
|     {1} | 0    0     0     0      5x_0  4x_1  6x_2   -28x_3 -7x_4 0     0   |
|     {1} | 0    0     0     0      0     x_0   8x_1   42x_2  0     -7x_4 0   |
|     {1} | 0    0     0     0      0     0     x_0    20x_1  6x_2  4x_3  -x_4|
|     {1} | 0    0     0     0      0     0     0      x_0    x_1   3x_2  x_3 |
|     ------------------------------------------------------------------------|
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|                                                                             |
|               ZZ          9         ZZ          11                          |
|o4 : Matrix (-----[x ..x ])  <--- (-----[x ..x ])                            |
|             10007  0   4          10007  0   4                              |
+-----------------------------------------------------------------------------+

If the first argument is a polynomial ring R, then d = numgens R-1.

+-----------------------------------------------------------------------------+
|i5 : R = QQ[y_0..y_4];                                                       |
+-----------------------------------------------------------------------------+
|i6 : m = 3                                                                   |
|                                                                             |
|o6 = 3                                                                       |
+-----------------------------------------------------------------------------+
|i7 : M = sl2EquivariantConstantRankMatrix(R,m)                               |
|                                                                             |
|o7 = {1} | -y_1 -3y_2 -y_3  -y_4   0     0     0      0      0     0     0   |
|     {1} | y_0  -4y_1 -6y_2 -20y_3 -y_4  0     0      0      0     0     0   |
|     {1} | 0    7y_0  0     -42y_2 -8y_3 -y_4  0      0      0     0     0   |
|     {1} | 0    0     7y_0  28y_1  -6y_2 -4y_3 -5y_4  0      0     0     0   |
|     {1} | 0    0     0     35y_0  10y_1 0     -10y_3 -35y_4 0     0     0   |
|     {1} | 0    0     0     0      5y_0  4y_1  6y_2   -28y_3 -7y_4 0     0   |
|     {1} | 0    0     0     0      0     y_0   8y_1   42y_2  0     -7y_4 0   |
|     {1} | 0    0     0     0      0     0     y_0    20y_1  6y_2  4y_3  -y_4|
|     {1} | 0    0     0     0      0     0     0      y_0    y_1   3y_2  y_3 |
|     ------------------------------------------------------------------------|
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|     |                                                                       |
|                                                                             |
|             9       11                                                      |
|o7 : Matrix R  <--- R                                                        |
+-----------------------------------------------------------------------------+

Ways to use sl2EquivariantConstantRankMatrix :
==============================================

  * "sl2EquivariantConstantRankMatrix(PolynomialRing,ZZ)"
  * "sl2EquivariantConstantRankMatrix(ZZ,ZZ)"

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

The object sl2EquivariantConstantRankMatrix,
is a method function with options (Macaulay2Doc).

sl2EquivariantConstantRankMatrix(...,CoefficientRing=>...) -- name for optional argument
****************************************************************************************

Synopsis
========

  * Usage:
        M = sl2EquivariantConstantRankMatrix(m,d,CoefficientRing=>C)

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

This is an option to tell slEquivariantConstantRankMatrix,
to define the matrix over a polynomial ring
with coefficients in the ring C.

Further information
===================

  * Default value: QQ (Macaulay2Doc),
  * Function: sl2EquivariantConstantRankMatrix,
    -- computes a SL(2)-equivariant constant
    rank matrix
  * Option key: CoefficientRing (Macaulay2Doc), -- an
    optional argument

Functions with optional argument named CoefficientRing :
========================================================

  * flattenRing(...,CoefficientRing=>...) (Macaulay2Doc),
    --
    optionally specify the desired coefficient ring of the flattened ring
  * "generators(...,CoefficientRing=>...)" -- see generators(Ring) (Macaulay2Doc),
    -- the list of generators of a ring
  * "Grassmannian(...,CoefficientRing=>...)" -- see Grassmannian(ZZ,ZZ) (Macaulay2Doc),
    -- the Grassmannian of linear
    subspaces of a vector space
  * "Schubert(...,CoefficientRing=>...)" -- see
    Schubert(ZZ,ZZ,VisibleList) (Macaulay2Doc),
    -- find the Plücker
    ideal of a Schubert variety
  * sl2EquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * sl2EquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument
  * slEquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * slEquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument

sl2EquivariantVectorBundle -- computes a SL(2)-equivariant vector bundle over some projective space
***************************************************************************************************

Synopsis
========

  * Usage:
        W = sl2EquivariantVectorBundle(d,m)
        W = sl2EquivariantVectorBundle(d,m,CoefficientRing=>C)
        W = sl2EquivariantVectorBundle(R,m)
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), with at
        least two variables
      * d, an integer (Macaulay2Doc), positive
      * m, an integer (Macaulay2Doc), at least 2
      * C, a ring (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing,
        => ..., default value QQ, name for optional argument
  * Outputs:
      * W, a coherent sheaf (Macaulay2Doc), vector bundle

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

This function returns the kernel of the matrix describing the morphism

$\Phi: S^{md-2}V \otimes O_{\PP^d} \to S^{(m-1)d}V \otimes O_{\PP^d)}(1)$

given by the projection

$S^dV \otimes S^{(m-1)d}V \to S^{md-2}V$

of the irreducible $SL(2)$-subrepresentation of highest weight $md-2$, where
$\PP^d = \PP(S^dV)$ as $V=<v_0,v_1>$.

In the paper A construction of equivariant bundles on the space of symmetric
forms, it is proved that the matrix $\Phi$ has constant co-rank 1, so that the
kernel $W = ker \Phi$ turns out to be a vector bundle, and the entries of the
matrix $\Phi$ are explicitly described.

+----------------------------------------------------------------------------+
|i1 : d = 3, m = 2                                                           |
|                                                                            |
|o1 = (3, 2)                                                                 |
|                                                                            |
|o1 : Sequence                                                               |
+----------------------------------------------------------------------------+
|i2 : W = sl2EquivariantVectorBundle(d,m)                                    |
|                                                                            |
|o2 = cokernel {4} | 0    x_3   0     x_2  |                                 |
|              {4} | x_1  0     x_0   0    |                                 |
|              {4} | -x_2 x_0   0     0    |                                 |
|              {4} | x_3  -3x_1 -3x_2 x_0  |                                 |
|              {4} | 0    0     x_3   -x_1 |                                 |
|                                                                            |
|                                                                       5    |
|o2 : coherent sheaf on Proj(QQ[x ..x ]), quotient of OO                 (-4)|
|                                0   3                  Proj(QQ[x ..x ])     |
|                                                                0   3       |
+----------------------------------------------------------------------------+

By default, see slEquivariantVectorBundle, defines
the vector bundle over a projective space whose coordinate ring has rational
coefficients. The optional argument CoefficientRing (Macaulay2Doc),
allows one to change the coefficient ring.

+-----------------------------------------------------------------------------+
|i3 : d = 3, m = 2                                                            |
|                                                                             |
|o3 = (3, 2)                                                                  |
|                                                                             |
|o3 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i4 : W = sl2EquivariantVectorBundle(d,m,CoefficientRing=>ZZ/10007)           |
|                                                                             |
|o4 = cokernel {4} | 0       3336x_3 0       x_2  |                           |
|              {4} | x_1     0       x_0     0    |                           |
|              {4} | -x_2    x_0     0       0    |                           |
|              {4} | 3336x_3 -x_1    -x_2    x_0  |                           |
|              {4} | 0       0       3336x_3 -x_1 |                           |
|                                                                             |
|                              ZZ                                             |
|o4 : coherent sheaf on Proj(-----[x ..x ]), quotient of OO                   |
|                            10007  0   3                         ZZ          |
|                                                          Proj(-----[x ..x ])|
|                                                               10007  0   3  |
|-----------------------------------------------------------------------------|
|5                                                                            |
| (-4)                                                                        |
+-----------------------------------------------------------------------------+

If the first argument is a polynomial ring R, then d = numgens R-1.

+--------------------------------------------------------+
|i5 : R = QQ[y_0..y_3];                                  |
+--------------------------------------------------------+
|i6 : m = 2                                              |
|                                                        |
|o6 = 2                                                  |
+--------------------------------------------------------+
|i7 : W = sl2EquivariantVectorBundle(R,m)                |
|                                                        |
|o7 = cokernel {4} | 0    y_3   0     y_2  |             |
|              {4} | y_1  0     y_0   0    |             |
|              {4} | -y_2 y_0   0     0    |             |
|              {4} | y_3  -3y_1 -3y_2 y_0  |             |
|              {4} | 0    0     y_3   -y_1 |             |
|                                                        |
|                                                   5    |
|o7 : coherent sheaf on Proj R, quotient of OO       (-4)|
|                                             Proj R     |
+--------------------------------------------------------+

Ways to use sl2EquivariantVectorBundle :
========================================

  * "sl2EquivariantVectorBundle(PolynomialRing,ZZ)"
  * "sl2EquivariantVectorBundle(ZZ,ZZ)"

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

The object sl2EquivariantVectorBundle, is a
method function with options (Macaulay2Doc).

sl2EquivariantVectorBundle(...,CoefficientRing=>...) -- name for optional argument
**********************************************************************************

Synopsis
========

  * Usage:
        W = sl2EquivariantVectorBundle(m,d,CoefficientRing=>C)

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

This is an option to tell slEquivariantVectorBundle,
to define the vector bundle over a projective space
whose coordinate ring has coefficients in the ring C.

Further information
===================

  * Default value: QQ (Macaulay2Doc),
  * Function: sl2EquivariantVectorBundle, --
    computes a SL(2)-equivariant vector bundle over some projective space
  * Option key: CoefficientRing (Macaulay2Doc), -- an
    optional argument

Functions with optional argument named CoefficientRing :
========================================================

  * flattenRing(...,CoefficientRing=>...) (Macaulay2Doc),
    --
    optionally specify the desired coefficient ring of the flattened ring
  * "generators(...,CoefficientRing=>...)" -- see generators(Ring) (Macaulay2Doc),
    -- the list of generators of a ring
  * "Grassmannian(...,CoefficientRing=>...)" -- see Grassmannian(ZZ,ZZ) (Macaulay2Doc),
    -- the Grassmannian of linear
    subspaces of a vector space
  * "Schubert(...,CoefficientRing=>...)" -- see
    Schubert(ZZ,ZZ,VisibleList) (Macaulay2Doc),
    -- find the Plücker
    ideal of a Schubert variety
  * sl2EquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * sl2EquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument
  * slEquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * slEquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument

slEquivariantConstantRankMatrix -- computes a SL-equivariant constant rank matrix
*********************************************************************************

Synopsis
========

  * Usage:
        M = slEquivariantConstantRankMatrix(n,d,m)
        M = slEquivariantConstantRankMatrix(n,d,m,CoefficientRing=>C)
        M = slEquivariantConstantRankMatrix(R,d,m)
        M = slEquivariantConstantRankMatrix(n,d,m,X)
        M = slEquivariantConstantRankMatrix(R,d,m,X)
  * Inputs:
      * n, an integer (Macaulay2Doc), positive
      * R, a polynomial ring (Macaulay2Doc), with at
        least two variables
      * d, an integer (Macaulay2Doc), positive
      * m, an integer (Macaulay2Doc), at least 2
      * X, a polynomial ring (Macaulay2Doc), with
        binomial(n+d,n) or binomial(numgens R-1+d,numgens R-1) variables
      * C, a ring (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing,
        => ..., default value QQ, name for optional argument
  * Outputs:
      * M, a matrix (Macaulay2Doc),

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

This function returns a constant rank matrix of linear forms. For $n=1$, the
matrix describes the morphism

$\Phi: S^{md-2}V \otimes O_{\PP^d} \to S^{(m-1)d}V \otimes O_{\PP^d)}(1)$

given by the projection

$S^dV \otimes S^{(m-1)d}V \to S^{md-2}V$

of the irreducible $SL(2)$-subrepresentation of highest weight $md-2$, where
$\PP^d = \PP(S^dV)$ as $V=<v_0,v_1>$.

For $n>1$, the matrix describes the morphism

$\Phi: V_{(md-2)\lambda_1 + \lambda_2} \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V
\otimes O_{\PP(S^dV)}(1)$

given by the projection

$S^dV \otimes S^{(m-1)d}V \to V_{(md-2)\lambda_1 + \lambda_2}$

of the irreducible $SL(n+1)$-subrepresentation $V_{(md-2)\lambda_1 +
\lambda_2}$ of highest weight $(md-2)\lambda_1 + \lambda_2 = (md-1)L_1 + L_2$
in the tensor product $S^dV \otimes S^{(m-1)d}V$, where $V = \CC^{n+1}$ and
$\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of the Lie
group $SL(n+1)$.

+------------------------------------------------------------+
|i1 : n = 1, d = 3, m = 3                                    |
|                                                            |
|o1 = (1, 3, 3)                                              |
|                                                            |
|o1 : Sequence                                               |
+------------------------------------------------------------+
|i2 : M = slEquivariantConstantRankMatrix(n,d,m)             |
|                                                            |
|o2 = {-1} | -x_1 2x_2  x_3    0     0     0     0     0    ||
|     {-1} | x_0  3x_1  9x_2   x_3   0     0     0     0    ||
|     {-1} | 0    -5x_0 0      3x_2  2x_3  0     0     0    ||
|     {-1} | 0    0     -10x_0 -2x_1 2x_2  10x_3 0     0    ||
|     {-1} | 0    0     0      -2x_0 -3x_1 0     5x_3  0    ||
|     {-1} | 0    0     0      0     -x_0  -9x_1 -3x_2 x_3  ||
|     {-1} | 0    0     0      0     0     -x_0  -2x_1 -x_2 ||
|                                                            |
|                        7                  8                |
|o2 : Matrix (QQ[x ..x ])  <--- (QQ[x ..x ])                 |
|                 0   3              0   3                   |
+------------------------------------------------------------+

By default, see slEquivariantConstantRankMatrix,
defines the matrix over a polynomial ring with
rational coefficients. The optional argument CoefficientRing (Macaulay2Doc),
allows one to change the coefficient ring.

+-------------------------------------------------------------------------+
|i3 : n = 1, d = 3, m = 3                                                 |
|                                                                         |
|o3 = (1, 3, 3)                                                           |
|                                                                         |
|o3 : Sequence                                                            |
+-------------------------------------------------------------------------+
|i4 : M = slEquivariantConstantRankMatrix(n,d,m,CoefficientRing=>ZZ/10007)|
|                                                                         |
|o4 = {-1} | -x_1 2x_2  x_3    0     0     0     0     0    |             |
|     {-1} | x_0  3x_1  9x_2   x_3   0     0     0     0    |             |
|     {-1} | 0    -5x_0 0      3x_2  2x_3  0     0     0    |             |
|     {-1} | 0    0     -10x_0 -2x_1 2x_2  10x_3 0     0    |             |
|     {-1} | 0    0     0      -2x_0 -3x_1 0     5x_3  0    |             |
|     {-1} | 0    0     0      0     -x_0  -9x_1 -3x_2 x_3  |             |
|     {-1} | 0    0     0      0     0     -x_0  -2x_1 -x_2 |             |
|                                                                         |
|               ZZ          7         ZZ          8                       |
|o4 : Matrix (-----[x ..x ])  <--- (-----[x ..x ])                        |
|             10007  0   3          10007  0   3                          |
+-------------------------------------------------------------------------+

If the first argument is a polynomial ring R, then n = numgens R-1.

+-----------------------------------------------+
|i5 : R = QQ[y_0,y_1];                          |
+-----------------------------------------------+
|i6 : d = 2, m = 3                              |
|                                               |
|o6 = (2, 3)                                    |
|                                               |
|o6 : Sequence                                  |
+-----------------------------------------------+
|i7 : M = slEquivariantConstantRankMatrix(R,d,m)|
|                                               |
|o7 = {-1} | -x_1 x_2   0    0     0    |       |
|     {-1} | x_0  2x_1  x_2  0     0    |       |
|     {-1} | 0    -3x_0 0    3x_2  0    |       |
|     {-1} | 0    0     -x_0 -2x_1 x_2  |       |
|     {-1} | 0    0     0    -x_0  -x_1 |       |
|                                               |
|                        5                  5   |
|o7 : Matrix (QQ[x ..x ])  <--- (QQ[x ..x ])    |
|                 0   2              0   2      |
+-----------------------------------------------+

If the last argument is polynomial ring X (and X has the same number of
variables of the coordinate ring of $\PP(S^d\CC^{n+1})$), then the matrix is
defined over the polynomial ring X.

+-----------------------------------------------------------+
|i8 : n = 1, d = 3, m = 3                                   |
|                                                           |
|o8 = (1, 3, 3)                                             |
|                                                           |
|o8 : Sequence                                              |
+-----------------------------------------------------------+
|i9 : X = ZZ/7[z_0,z_1,z_2,z_3];                            |
+-----------------------------------------------------------+
|i10 : M = slEquivariantConstantRankMatrix(n,d,m,X)         |
|                                                           |
|o10 = {-1} | -z_1 2z_2 z_3   0     0     0     0     0    ||
|      {-1} | z_0  3z_1 2z_2  z_3   0     0     0     0    ||
|      {-1} | 0    2z_0 0     3z_2  2z_3  0     0     0    ||
|      {-1} | 0    0    -3z_0 -2z_1 2z_2  3z_3  0     0    ||
|      {-1} | 0    0    0     -2z_0 -3z_1 0     -2z_3 0    ||
|      {-1} | 0    0    0     0     -z_0  -2z_1 -3z_2 z_3  ||
|      {-1} | 0    0    0     0     0     -z_0  -2z_1 -z_2 ||
|                                                           |
|              7       8                                    |
|o10 : Matrix X  <--- X                                     |
+-----------------------------------------------------------+
|i11 : R = QQ[y_0,y_1];                                     |
+-----------------------------------------------------------+
|i12 : d = 3, m = 2                                         |
|                                                           |
|o12 = (3, 2)                                               |
|                                                           |
|o12 : Sequence                                             |
+-----------------------------------------------------------+
|i13 : M = slEquivariantConstantRankMatrix(R,d,m,X)         |
|                                                           |
|o13 = {-1} | -z_1 z_2  z_3   0    0    |                   |
|      {-1} | z_0  0    3z_2  z_3  0    |                   |
|      {-1} | 0    -z_0 -3z_1 0    z_3  |                   |
|      {-1} | 0    0    -z_0  -z_1 -z_2 |                   |
|                                                           |
|              4       5                                    |
|o13 : Matrix X  <--- X                                     |
+-----------------------------------------------------------+

Ways to use slEquivariantConstantRankMatrix :
=============================================

  * "slEquivariantConstantRankMatrix(PolynomialRing,ZZ,ZZ)"
  * "slEquivariantConstantRankMatrix(PolynomialRing,ZZ,ZZ,PolynomialRing)"
  * "slEquivariantConstantRankMatrix(ZZ,ZZ,ZZ)"
  * "slEquivariantConstantRankMatrix(ZZ,ZZ,ZZ,PolynomialRing)"

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

The object slEquivariantConstantRankMatrix,
is a method function with options (Macaulay2Doc).

slEquivariantConstantRankMatrix(...,CoefficientRing=>...) -- name for optional argument
***************************************************************************************

Synopsis
========

  * Usage:
        M = slEquivariantConstantRankMatrix(n,m,d,CoefficientRing=>C)

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

This is an option to tell slEquivariantConstantRankMatrix,
to define the matrix over a polynomial ring
with coefficients in the ring C.

Further information
===================

  * Default value: QQ (Macaulay2Doc),
  * Function: slEquivariantConstantRankMatrix,
    -- computes a SL-equivariant constant rank
    matrix
  * Option key: CoefficientRing (Macaulay2Doc), -- an
    optional argument

Functions with optional argument named CoefficientRing :
========================================================

  * flattenRing(...,CoefficientRing=>...) (Macaulay2Doc),
    --
    optionally specify the desired coefficient ring of the flattened ring
  * "generators(...,CoefficientRing=>...)" -- see generators(Ring) (Macaulay2Doc),
    -- the list of generators of a ring
  * "Grassmannian(...,CoefficientRing=>...)" -- see Grassmannian(ZZ,ZZ) (Macaulay2Doc),
    -- the Grassmannian of linear
    subspaces of a vector space
  * "Schubert(...,CoefficientRing=>...)" -- see
    Schubert(ZZ,ZZ,VisibleList) (Macaulay2Doc),
    -- find the Plücker
    ideal of a Schubert variety
  * sl2EquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * sl2EquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument
  * slEquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * slEquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument

slEquivariantVectorBundle -- computes a SL-equivariant vector bundle over some projective space
***********************************************************************************************

Synopsis
========

  * Usage:
        W = slEquivariantVectorBundle(n,d,m)
        W = slEquivariantVectorBundle(n,d,m,CoefficientRing=>C)
        W = slEquivariantVectorBundle(R,d,m)
        W = slEquivariantVectorBundle(n,d,m,X)
        W = slEquivariantVectorBundle(R,d,m,X)
  * Inputs:
      * n, an integer (Macaulay2Doc), positive
      * R, a polynomial ring (Macaulay2Doc), with at
        least two variables
      * d, an integer (Macaulay2Doc), positive
      * m, an integer (Macaulay2Doc), at least 2
      * X, a polynomial ring (Macaulay2Doc), with
        binomial(n+d,n) or binomial(numgens R-1+d,numgens R-1) variables
      * C, a ring (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing,
        => ..., default value QQ, name for optional argument
  * Outputs:
      * W, a coherent sheaf (Macaulay2Doc), vector bundle

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

For $n=1$, this function returns the kernel of the matrix describing the
morphism

$\Phi: S^{md-2}V \otimes O_{\PP^d} \to S^{(m-1)d}V \otimes O_{\PP^d)}(1)$

given by the projection

$S^dV \otimes S^{(m-1)d}V \to S^{md-2}V$

of the irreducible $SL(2)$-subrepresentation of highest weight $md-2$, where
$\PP^d = \PP(S^dV)$ as $V=<v_0,v_1>$, while for $n>1$, the function returns the
kernel of the matrix describing the morphism

$\Phi: V_{(md-2)\lambda_1 + \lambda_2} \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V
\otimes O_{\PP(S^dV)}(1)$

given by the projection

$S^dV \otimes S^{(m-1)d}V \to V_{(md-2)\lambda_1 + \lambda_2}$

of the irreducible $SL(n+1)$-subrepresentation $V_{(md-2)\lambda_1 +
\lambda_2}$ of highest weight $(md-2)\lambda_1 + \lambda_2 = (md-1)L_1 + L_2$
in the tensor product $S^dV \otimes S^{(m-1)d}V$, where $V = \CC^{n+1}$ and
$\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of the Lie
group $SL(n+1)$.

In the paper A construction of equivariant bundles on the space of symmetric
forms, it is proved that the matrix $\Phi$ has constant co-rank 1, so that the
kernel $W = ker \Phi$ turns out to be a vector bundle.

+----------------------------------------------------------------------------+
|i1 : n = 1, d = 3, m = 3                                                    |
|                                                                            |
|o1 = (1, 3, 3)                                                              |
|                                                                            |
|o1 : Sequence                                                               |
+----------------------------------------------------------------------------+
|i2 : W = slEquivariantVectorBundle(n,d,m)                                   |
|                                                                            |
|o2 = cokernel {4} | 0     0     0     0    0     x_3   x_2   |              |
|              {4} | x_1   x_0   0     0    0     0     0     |              |
|              {4} | -2x_2 3x_1  x_0   0    0     0     0     |              |
|              {4} | x_3   -9x_2 0     x_0  0     0     0     |              |
|              {4} | 0     5x_3  -3x_2 -x_1 2x_0  0     0     |              |
|              {4} | 0     0     2x_3  -x_2 -3x_1 5x_0  0     |              |
|              {4} | 0     0     0     x_3  0     -9x_1 x_0   |              |
|              {4} | 0     0     0     0    x_3   3x_2  -2x_1 |              |
|                                                                            |
|                                                                       8    |
|o2 : coherent sheaf on Proj(QQ[x ..x ]), quotient of OO                 (-4)|
|                                0   3                  Proj(QQ[x ..x ])     |
|                                                                0   3       |
+----------------------------------------------------------------------------+

By default, see slEquivariantVectorBundle, defines
the vector bundle over a projective space whose coordinate ring has rational
coefficients. The optional argument CoefficientRing (Macaulay2Doc),
allows one to change the coefficient ring.

+-----------------------------------------------------------------------------+
|i3 : n = 1, d = 3, m = 3                                                     |
|                                                                             |
|o3 = (1, 3, 3)                                                               |
|                                                                             |
|o3 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i4 : W = slEquivariantVectorBundle(n,d,m,CoefficientRing=>ZZ/10007)          |
|                                                                             |
|o4 = cokernel {4} | 0       0        0       0       0        556x_3  -5003x_|
|              {4} | x_1     x_0      0       0       0        0       0      |
|              {4} | -x_2    -5002x_1 x_0     0       0        0       0      |
|              {4} | 1668x_3 5002x_2  0       x_0     0        0       0      |
|              {4} | 0       -1667x_3 -x_2    -x_1    x_0      0       0      |
|              {4} | 0       0        4448x_3 3335x_2 -x_1     x_0     0      |
|              {4} | 0       0        0       371x_3  0        -x_1    x_0    |
|              {4} | 0       0        0       0       -2409x_3 1668x_2 -x_1   |
|                                                                             |
|                              ZZ                                             |
|o4 : coherent sheaf on Proj(-----[x ..x ]), quotient of OO                   |
|                            10007  0   3                         ZZ          |
|                                                          Proj(-----[x ..x ])|
|                                                               10007  0   3  |
|-----------------------------------------------------------------------------|
|2 |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|                                                                             |
|8                                                                            |
| (-4)                                                                        |
+-----------------------------------------------------------------------------+

If the first argument is a polynomial ring R, then n = numgens R-1.

+---------------------------------------------+
|i5 : R = QQ[y_0,y_1];                        |
+---------------------------------------------+
|i6 : d = 2, m = 3                            |
|                                             |
|o6 = (2, 3)                                  |
|                                             |
|o6 : Sequence                                |
+---------------------------------------------+
|i7 : W = slEquivariantVectorBundle(R,d,m)    |
|                                             |
|                       1                     |
|o7 = OO                 (-2)                 |
|       Proj(QQ[x ..x ])                      |
|                0   2                        |
|                                             |
|o7 : coherent sheaf on Proj(QQ[x ..x ]), free|
|                                0   2        |
+---------------------------------------------+

If the last argument is polynomial ring X (and X has the same number of
variables of the coordinate ring of $\PP(S^d\CC^{n+1})$), then the vector
bundle is defined over the projective space Proj(X).

+------------------------------------------------------------+
|i8 : n = 1, d = 3, m = 3                                    |
|                                                            |
|o8 = (1, 3, 3)                                              |
|                                                            |
|o8 : Sequence                                               |
+------------------------------------------------------------+
|i9 : X = ZZ/7[z_0,z_1,z_2,z_3];                             |
+------------------------------------------------------------+
|i10 : W = slEquivariantVectorBundle(n,d,m,X)                |
|                                                            |
|o10 = cokernel {4} | 0    0     0    0     0    2z_3 -3z_2 ||
|               {4} | z_1  z_0   0    0     0    0    0     ||
|               {4} | -z_2 -2z_1 z_0  0     0    0    0     ||
|               {4} | -z_3 2z_2  0    z_0   0    0    0     ||
|               {4} | 0    2z_3  -z_2 -z_1  z_0  0    0     ||
|               {4} | 0    0     2z_3 -3z_2 -z_1 z_0  0     ||
|               {4} | 0    0     0    -3z_3 0    -z_1 z_0   ||
|               {4} | 0    0     0    0     z_3  -z_2 -z_1  ||
|                                                            |
|                                                    8       |
|o10 : coherent sheaf on Proj X, quotient of OO       (-4)   |
|                                              Proj X        |
+------------------------------------------------------------+
|i11 : R = QQ[y_0,y_1];                                      |
+------------------------------------------------------------+
|i12 : d = 3, m = 2                                          |
|                                                            |
|o12 = (3, 2)                                                |
|                                                            |
|o12 : Sequence                                              |
+------------------------------------------------------------+
|i13 : W = slEquivariantVectorBundle(R,d,m,X)                |
|                                                            |
|o13 = cokernel {2} | 0     -2z_3 0     z_2  |               |
|               {2} | z_1   0     z_0   0    |               |
|               {2} | -z_2  z_0   0     0    |               |
|               {2} | -2z_3 -z_1  -z_2  z_0  |               |
|               {2} | 0     0     -2z_3 -z_1 |               |
|                                                            |
|                                                    5       |
|o13 : coherent sheaf on Proj X, quotient of OO       (-2)   |
|                                              Proj X        |
+------------------------------------------------------------+

Ways to use slEquivariantVectorBundle :
=======================================

  * "slEquivariantVectorBundle(PolynomialRing,ZZ,ZZ)"
  * "slEquivariantVectorBundle(PolynomialRing,ZZ,ZZ,PolynomialRing)"
  * "slEquivariantVectorBundle(ZZ,ZZ,ZZ)"
  * "slEquivariantVectorBundle(ZZ,ZZ,ZZ,PolynomialRing)"

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

The object slEquivariantVectorBundle, is a
method function with options (Macaulay2Doc).

slEquivariantVectorBundle(...,CoefficientRing=>...) -- name for optional argument
*********************************************************************************

Synopsis
========

  * Usage:
        W = slEquivariantVectorBundle(n,m,d,CoefficientRing=>C)

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

This is an option to tell slEquivariantVectorBundle,
to define the vector bundle over a projective space
whose coordinate ring has coefficients in the ring C.

Further information
===================

  * Default value: QQ (Macaulay2Doc),
  * Function: slEquivariantVectorBundle, --
    computes a SL-equivariant vector bundle over some projective space
  * Option key: CoefficientRing (Macaulay2Doc), -- an
    optional argument

Functions with optional argument named CoefficientRing :
========================================================

  * flattenRing(...,CoefficientRing=>...) (Macaulay2Doc),
    --
    optionally specify the desired coefficient ring of the flattened ring
  * "generators(...,CoefficientRing=>...)" -- see generators(Ring) (Macaulay2Doc),
    -- the list of generators of a ring
  * "Grassmannian(...,CoefficientRing=>...)" -- see Grassmannian(ZZ,ZZ) (Macaulay2Doc),
    -- the Grassmannian of linear
    subspaces of a vector space
  * "Schubert(...,CoefficientRing=>...)" -- see
    Schubert(ZZ,ZZ,VisibleList) (Macaulay2Doc),
    -- find the Plücker
    ideal of a Schubert variety
  * sl2EquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * sl2EquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument
  * slEquivariantConstantRankMatrix(...,CoefficientRing=>...)
    -- name for optional argument
  * slEquivariantVectorBundle(...,CoefficientRing=>...),
    --
    name for optional argument

slIrreducibleRepresentationsTensorProduct -- computes the irreducible SL-subrepresentations of the tensor product of two symmetric products
*******************************************************************************************************************************************

Synopsis
========

  * Usage:
        D = slIrreducibleRepresentationsTensorProduct(n,a,b)
        D = slIrreducibleRepresentationsTensorProduct(R,a,b)
  * Inputs:
      * n, an integer (Macaulay2Doc), positive
      * R, a polynomial ring (Macaulay2Doc),
      * a, an integer (Macaulay2Doc), positive
      * b, an integer (Macaulay2Doc), positive
  * Outputs:
      * D, a list (Macaulay2Doc), of irreducible representations

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

This function computes the decomposition in irreducible
$SL(n+1)$-representations of the tensor product $S^aV \otimes S^bV$, where $V =
<v_0,\ldots,v_n>$ and $a \leq b$.

If $n = 1$, the decomposition is

$S^aV \otimes S^bV = S^{a+b}V \oplus S^{a+b-2}V \oplus S^{a+b-4}V \oplus \dots
\oplus S^{b-a}V$,

while if $n > 1$, the decomposition is

$S^aV \otimes S^bV = S^{a+b}V \oplus V_{(a+b-2)\lambda_1 + \lambda_2} \oplus
V_{(a+b-4)\lambda_1 + 2\lambda_2} \oplus \dots \oplus V_{(b-a)\lambda_1 +
a\lambda_2}$,

where $\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of
the Lie group $SL(n+1)$ and $V_{i\lambda_1+j\lambda_2}$ is the irreducible
representation of highest weight $i\lambda_1+j\lambda_2$.

+-----------------------------------------------------------------------------+
|i1 : n = 2                                                                   |
|                                                                             |
|o1 = 2                                                                       |
+-----------------------------------------------------------------------------+
|i2 : a = 1, b = 2                                                            |
|                                                                             |
|o2 = (1, 2)                                                                  |
|                                                                             |
|o2 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i3 : D = slIrreducibleRepresentationsTensorProduct(n,a,b);                   |
+-----------------------------------------------------------------------------+
|i4 : #D                                                                      |
|                                                                             |
|o4 = 2                                                                       |
+-----------------------------------------------------------------------------+
|i5 : D#0                                                                     |
|                                                                             |
|           2                         2         2                             |
|o5 = {v   v   , 2v   v   v    + v   v   , v   v    + 2v   v   v   ,          |
|       0,0 0,1    0,0 0,1 1,1    1,0 0,1   0,0 1,1     1,0 0,1 1,1           |
|     ------------------------------------------------------------------------|
|                          2         2                                        |
|     2v   v   v    + v   v   , v   v   , v   v   v    + v   v   v    +       |
|       0,0 0,1 2,1    2,0 0,1   1,0 1,1   0,0 1,1 2,1    1,0 0,1 2,1         |
|     ------------------------------------------------------------------------|
|                                        2         2                          |
|     v   v   v   , 2v   v   v    + v   v   , v   v    + 2v   v   v   ,       |
|      2,0 0,1 1,1    1,0 1,1 2,1    2,0 1,1   0,0 2,1     2,0 0,1 2,1        |
|     ------------------------------------------------------------------------|
|          2                         2                                        |
|     v   v    + 2v   v   v   , v   v   }                                     |
|      1,0 2,1     2,0 1,1 2,1   2,0 2,1                                      |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+
|i6 : D#1                                                                     |
|                                                                             |
|                          2         2                                        |
|o6 = {v   v   v    - v   v   , v   v    - v   v   v   , v   v   v    -       |
|       0,0 0,1 1,1    1,0 0,1   0,0 1,1    1,0 0,1 1,1   0,0 0,1 2,1         |
|     ------------------------------------------------------------------------|
|          2                                                                  |
|     v   v   , v   v   v    - v   v   v   , v   v   v    - v   v   v   ,     |
|      2,0 0,1   0,0 1,1 2,1    2,0 0,1 1,1   1,0 0,1 2,1    2,0 0,1 1,1      |
|     ------------------------------------------------------------------------|
|                         2         2                        2                |
|     v   v   v    - v   v   , v   v    - v   v   v   , v   v    -            |
|      1,0 1,1 2,1    2,0 1,1   0,0 2,1    2,0 0,1 2,1   1,0 2,1              |
|     ------------------------------------------------------------------------|
|     v   v   v   }                                                           |
|      2,0 1,1 2,1                                                            |
|                                                                             |
|o6 : List                                                                    |
+-----------------------------------------------------------------------------+

If a polynomial ring R is given, then n = numgens R - 1 and $V =
<R_0,\ldots,R_n>$.

+-----------------------------------------------------------------------------+
|i7 : R = QQ[x_0,x_1,x_2];                                                    |
+-----------------------------------------------------------------------------+
|i8 : a = 2, b = 3                                                            |
|                                                                             |
|o8 = (2, 3)                                                                  |
|                                                                             |
|o8 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i9 : D = slIrreducibleRepresentationsTensorProduct(R,a,b);                   |
+-----------------------------------------------------------------------------+
|i10 : #D                                                                     |
|                                                                             |
|o10 = 3                                                                      |
+-----------------------------------------------------------------------------+
|i11 : D#0                                                                    |
|                                                                             |
|        2   3      2   2                   3      2       2                  |
|o11 = {x   x   , 3x   x   x    + 2x   x   x   , 3x   x   x    +              |
|        0,0 0,1    0,0 0,1 1,1     0,0 1,0 0,1    0,0 0,1 1,1                |
|      -----------------------------------------------------------------------|
|                2          2   3      2   2                   3     2   3    |
|      6x   x   x   x    + x   x   , 3x   x   x    + 2x   x   x   , x   x    +|
|        0,0 1,0 0,1 1,1    1,0 0,1    0,0 0,1 2,1     0,0 2,0 0,1   0,0 1,1  |
|      -----------------------------------------------------------------------|
|                    2       2   2          2                                 |
|      6x   x   x   x    + 3x   x   x   , 3x   x   x   x    +                 |
|        0,0 1,0 0,1 1,1     1,0 0,1 1,1    0,0 0,1 1,1 2,1                   |
|      -----------------------------------------------------------------------|
|                2                   2                  3              3      |
|      3x   x   x   x    + 3x   x   x   x    + x   x   x   , 2x   x   x    +  |
|        0,0 1,0 0,1 2,1     0,0 2,0 0,1 1,1    1,0 2,0 0,1    0,0 1,0 1,1    |
|      -----------------------------------------------------------------------|
|        2       2     2   2                                  2   2           |
|      3x   x   x   , x   x   x    + 4x   x   x   x   x    + x   x   x    +   |
|        1,0 0,1 1,1   0,0 1,1 2,1     0,0 1,0 0,1 1,1 2,1    1,0 0,1 2,1     |
|      -----------------------------------------------------------------------|
|                    2               2          2       2                     |
|      2x   x   x   x    + 2x   x   x   x   , 3x   x   x    +                 |
|        0,0 2,0 0,1 1,1     1,0 2,0 0,1 1,1    0,0 0,1 2,1                   |
|      -----------------------------------------------------------------------|
|                2          2   3     2   3              2                    |
|      6x   x   x   x    + x   x   , x   x   , 3x   x   x   x    +            |
|        0,0 2,0 0,1 2,1    2,0 0,1   1,0 1,1    0,0 1,0 1,1 2,1              |
|      -----------------------------------------------------------------------|
|        2                          3                   2     2       2       |
|      3x   x   x   x    + x   x   x    + 3x   x   x   x   , x   x   x    +   |
|        1,0 0,1 1,1 2,1    0,0 2,0 1,1     1,0 2,0 0,1 1,1   0,0 1,1 2,1     |
|      -----------------------------------------------------------------------|
|                    2                                       2                |
|      2x   x   x   x    + 4x   x   x   x   x    + 2x   x   x   x    +        |
|        0,0 1,0 0,1 2,1     0,0 2,0 0,1 1,1 2,1     1,0 2,0 0,1 2,1          |
|      -----------------------------------------------------------------------|
|       2   2          2   2                   3                  2           |
|      x   x   x   , 3x   x   x    + 2x   x   x   , 2x   x   x   x    +       |
|       2,0 0,1 1,1    1,0 1,1 2,1     1,0 2,0 1,1    0,0 1,0 1,1 2,1         |
|      -----------------------------------------------------------------------|
|       2       2               2                                             |
|      x   x   x    + 2x   x   x   x    + 4x   x   x   x   x    +             |
|       1,0 0,1 2,1     0,0 2,0 1,1 2,1     1,0 2,0 0,1 1,1 2,1               |
|      -----------------------------------------------------------------------|
|       2       2     2   3                   2       2   2                   |
|      x   x   x   , x   x    + 6x   x   x   x    + 3x   x   x   ,            |
|       2,0 0,1 1,1   0,0 2,1     0,0 2,0 0,1 2,1     2,0 0,1 2,1             |
|      -----------------------------------------------------------------------|
|        2       2               2          2   3             3               |
|      3x   x   x    + 6x   x   x   x    + x   x   , x   x   x    +           |
|        1,0 1,1 2,1     1,0 2,0 1,1 2,1    2,0 1,1   0,0 1,0 2,1             |
|      -----------------------------------------------------------------------|
|                    2                   2       2                 2   3      |
|      3x   x   x   x    + 3x   x   x   x    + 3x   x   x   x   , x   x    +  |
|        0,0 2,0 1,1 2,1     1,0 2,0 0,1 2,1     2,0 0,1 1,1 2,1   1,0 2,1    |
|      -----------------------------------------------------------------------|
|                    2       2   2                  3       2       2         |
|      6x   x   x   x    + 3x   x   x   , 2x   x   x    + 3x   x   x   ,      |
|        1,0 2,0 1,1 2,1     2,0 1,1 2,1    0,0 2,0 2,1     2,0 0,1 2,1       |
|      -----------------------------------------------------------------------|
|                3       2       2     2   3                                  |
|      2x   x   x    + 3x   x   x   , x   x   }                               |
|        1,0 2,0 2,1     2,0 1,1 2,1   2,0 2,1                                |
|                                                                             |
|o11 : List                                                                   |
+-----------------------------------------------------------------------------+
|i12 : D#1                                                                    |
|                                                                             |
|        2   2                  3      2       2              2               |
|o12 = {x   x   x    - x   x   x   , 2x   x   x    - x   x   x   x    -       |
|        0,0 0,1 1,1    0,0 1,0 0,1    0,0 0,1 1,1    0,0 1,0 0,1 1,1         |
|      -----------------------------------------------------------------------|
|       2   3     2   2                  3     2   3                  2       |
|      x   x   , x   x   x    - x   x   x   , x   x    + x   x   x   x    -   |
|       1,0 0,1   0,0 0,1 2,1    0,0 2,0 0,1   0,0 1,1    0,0 1,0 0,1 1,1     |
|      -----------------------------------------------------------------------|
|        2   2          2                          2                  3       |
|      2x   x   x   , 2x   x   x   x    - x   x   x   x    - x   x   x   ,    |
|        1,0 0,1 1,1    0,0 0,1 1,1 2,1    0,0 2,0 0,1 1,1    1,0 2,0 0,1     |
|      -----------------------------------------------------------------------|
|               2                  2                 3      2       2         |
|      x   x   x   x    - x   x   x   x   , x   x   x    - x   x   x   ,      |
|       0,0 1,0 0,1 2,1    0,0 2,0 0,1 1,1   0,0 1,0 1,1    1,0 0,1 1,1       |
|      -----------------------------------------------------------------------|
|       2   2          2   2                      2              2            |
|      x   x   x    - x   x   x    + x   x   x   x    - x   x   x   x   ,     |
|       0,0 1,1 2,1    1,0 0,1 2,1    0,0 2,0 0,1 1,1    1,0 2,0 0,1 1,1      |
|      -----------------------------------------------------------------------|
|                               2   2                       2                 |
|      2x   x   x   x   x    + x   x   x    - 2x   x   x   x    -             |
|        0,0 1,0 0,1 1,1 2,1    1,0 0,1 2,1     0,0 2,0 0,1 1,1               |
|      -----------------------------------------------------------------------|
|               2          2       2              2          2   3            |
|      x   x   x   x   , 2x   x   x    - x   x   x   x    - x   x   ,         |
|       1,0 2,0 0,1 1,1    0,0 0,1 2,1    0,0 2,0 0,1 2,1    2,0 0,1          |
|      -----------------------------------------------------------------------|
|               2                      2      2                          3    |
|      x   x   x   x    - x   x   x   x   , 2x   x   x   x    - x   x   x    -|
|       0,0 1,0 1,1 2,1    1,0 2,0 0,1 1,1    1,0 0,1 1,1 2,1    0,0 2,0 1,1  |
|      -----------------------------------------------------------------------|
|                   2      2       2                                          |
|      x   x   x   x   , 2x   x   x    + 2x   x   x   x   x    -              |
|       1,0 2,0 0,1 1,1    0,0 1,1 2,1     0,0 2,0 0,1 1,1 2,1                |
|      -----------------------------------------------------------------------|
|                2          2   2                      2                      |
|      3x   x   x   x    - x   x   x   , 2x   x   x   x    -                  |
|        1,0 2,0 0,1 2,1    2,0 0,1 1,1    0,0 1,0 0,1 2,1                    |
|      -----------------------------------------------------------------------|
|                                       2          2   2         2   2        |
|      2x   x   x   x   x    + x   x   x   x    - x   x   x   , x   x   x    -|
|        0,0 2,0 0,1 1,1 2,1    1,0 2,0 0,1 2,1    2,0 0,1 1,1   1,0 1,1 2,1  |
|      -----------------------------------------------------------------------|
|               3                  2              2                           |
|      x   x   x   , 2x   x   x   x    + x   x   x   x    -                   |
|       1,0 2,0 1,1    0,0 1,0 1,1 2,1    0,0 2,0 1,1 2,1                     |
|      -----------------------------------------------------------------------|
|                               2       2      2       2               2      |
|      2x   x   x   x   x    - x   x   x   , 2x   x   x    - 3x   x   x   x   |
|        1,0 2,0 0,1 1,1 2,1    2,0 0,1 1,1    1,0 0,1 2,1     0,0 2,0 1,1 2,1|
|      -----------------------------------------------------------------------|
|                                 2       2     2   3                  2      |
|      + 2x   x   x   x   x    - x   x   x   , x   x    + x   x   x   x    -  |
|          1,0 2,0 0,1 1,1 2,1    2,0 0,1 1,1   0,0 2,1    0,0 2,0 0,1 2,1    |
|      -----------------------------------------------------------------------|
|        2   2          2       2              2          2   3               |
|      2x   x   x   , 2x   x   x    - x   x   x   x    - x   x   ,            |
|        2,0 0,1 2,1    1,0 1,1 2,1    1,0 2,0 1,1 2,1    2,0 1,1             |
|      -----------------------------------------------------------------------|
|               3                  2       2                             2    |
|      x   x   x    + x   x   x   x    - 2x   x   x   x   , x   x   x   x    -|
|       0,0 1,0 2,1    1,0 2,0 0,1 2,1     2,0 0,1 1,1 2,1   0,0 2,0 1,1 2,1  |
|      -----------------------------------------------------------------------|
|                   2     2   3                  2       2   2                |
|      x   x   x   x   , x   x    + x   x   x   x    - 2x   x   x   ,         |
|       1,0 2,0 0,1 2,1   1,0 2,1    1,0 2,0 1,1 2,1     2,0 1,1 2,1          |
|      -----------------------------------------------------------------------|
|               3      2       2             3      2       2                 |
|      x   x   x    - x   x   x   , x   x   x    - x   x   x   }              |
|       0,0 2,0 2,1    2,0 0,1 2,1   1,0 2,0 2,1    2,0 1,1 2,1               |
|                                                                             |
|o12 : List                                                                   |
+-----------------------------------------------------------------------------+
|i13 : D#2                                                                    |
|                                                                             |
|        2       2               2          2   3     2   3                   |
|o13 = {x   x   x    - 2x   x   x   x    + x   x   , x   x    -               |
|        0,0 0,1 1,1     0,0 1,0 0,1 1,1    1,0 0,1   0,0 1,1                 |
|      -----------------------------------------------------------------------|
|                    2      2   2         2                          2        |
|      2x   x   x   x    + x   x   x   , x   x   x   x    - x   x   x   x    -|
|        0,0 1,0 0,1 1,1    1,0 0,1 1,1   0,0 0,1 1,1 2,1    0,0 1,0 0,1 2,1  |
|      -----------------------------------------------------------------------|
|               2                  3     2   2          2   2                 |
|      x   x   x   x    + x   x   x   , x   x   x    - x   x   x    -         |
|       0,0 2,0 0,1 1,1    1,0 2,0 0,1   0,0 1,1 2,1    1,0 0,1 2,1           |
|      -----------------------------------------------------------------------|
|                    2               2                                        |
|      2x   x   x   x    + 2x   x   x   x   , x   x   x   x   x    -          |
|        0,0 2,0 0,1 1,1     1,0 2,0 0,1 1,1   0,0 1,0 0,1 1,1 2,1            |
|      -----------------------------------------------------------------------|
|       2   2                      2              2         2       2         |
|      x   x   x    - x   x   x   x    + x   x   x   x   , x   x   x    -     |
|       1,0 0,1 2,1    0,0 2,0 0,1 1,1    1,0 2,0 0,1 1,1   0,0 0,1 2,1       |
|      -----------------------------------------------------------------------|
|                2          2   3             2          2                    |
|      2x   x   x   x    + x   x   , x   x   x   x    - x   x   x   x    -    |
|        0,0 2,0 0,1 2,1    2,0 0,1   0,0 1,0 1,1 2,1    1,0 0,1 1,1 2,1      |
|      -----------------------------------------------------------------------|
|               3                  2     2       2                            |
|      x   x   x    + x   x   x   x   , x   x   x    - 2x   x   x   x   x    +|
|       0,0 2,0 1,1    1,0 2,0 0,1 1,1   0,0 1,1 2,1     0,0 2,0 0,1 1,1 2,1  |
|      -----------------------------------------------------------------------|
|       2   2                     2                                           |
|      x   x   x   , x   x   x   x    - x   x   x   x   x    -                |
|       2,0 0,1 1,1   0,0 1,0 0,1 2,1    0,0 2,0 0,1 1,1 2,1                  |
|      -----------------------------------------------------------------------|
|               2          2   2                     2              2         |
|      x   x   x   x    + x   x   x   , x   x   x   x    - x   x   x   x    - |
|       1,0 2,0 0,1 2,1    2,0 0,1 1,1   0,0 1,0 1,1 2,1    0,0 2,0 1,1 2,1   |
|      -----------------------------------------------------------------------|
|                              2       2     2       2                        |
|      x   x   x   x   x    + x   x   x   , x   x   x    -                    |
|       1,0 2,0 0,1 1,1 2,1    2,0 0,1 1,1   1,0 0,1 2,1                      |
|      -----------------------------------------------------------------------|
|                               2       2     2   3                   2       |
|      2x   x   x   x   x    + x   x   x   , x   x    - 2x   x   x   x    +   |
|        1,0 2,0 0,1 1,1 2,1    2,0 0,1 1,1   0,0 2,1     0,0 2,0 0,1 2,1     |
|      -----------------------------------------------------------------------|
|       2   2         2       2               2          2   3             3  |
|      x   x   x   , x   x   x    - 2x   x   x   x    + x   x   , x   x   x   |
|       2,0 0,1 2,1   1,0 1,1 2,1     1,0 2,0 1,1 2,1    2,0 1,1   0,0 1,0 2,1|
|      -----------------------------------------------------------------------|
|                     2                  2      2                 2   3       |
|      - x   x   x   x    - x   x   x   x    + x   x   x   x   , x   x    -   |
|         0,0 2,0 1,1 2,1    1,0 2,0 0,1 2,1    2,0 0,1 1,1 2,1   1,0 2,1     |
|      -----------------------------------------------------------------------|
|                    2      2   2                                             |
|      2x   x   x   x    + x   x   x   }                                      |
|        1,0 2,0 1,1 2,1    2,0 1,1 2,1                                       |
|                                                                             |
|o13 : List                                                                   |
+-----------------------------------------------------------------------------+

Ways to use slIrreducibleRepresentationsTensorProduct :
=======================================================

  * "slIrreducibleRepresentationsTensorProduct(PolynomialRing,ZZ,ZZ)"
  * "slIrreducibleRepresentationsTensorProduct(ZZ,ZZ,ZZ)"

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

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