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slEquivariantConstantRankMatrix -- computes a SL-equivariant constant rank matrix

Synopsis

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 :

For the programmer

The object slEquivariantConstantRankMatrix is a method function with options.