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

Synopsis

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 
     ------------------------------------------------------------------------
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                        9                 11
o2 : Matrix (QQ[x ..x ])  <-- (QQ[x ..x ])
                 0   4             0   4

By default, sl2EquivariantConstantRankMatrix defines the matrix over a polynomial ring with rational coefficients. The optional argument CoefficientRing 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 
     ------------------------------------------------------------------------
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               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 
     ------------------------------------------------------------------------
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             9      11
o7 : Matrix R  <-- R

Ways to use sl2EquivariantConstantRankMatrix :

For the programmer

The object sl2EquivariantConstantRankMatrix is a method function with options.