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

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

## 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, slEquivariantConstantRankMatrix defines the matrix over a polynomial ring with rational coefficients. The optional argument CoefficientRing 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SLnEquivariantMatrices.m2:686:0.