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 explicitely 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, 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 | 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 |
The object sl2EquivariantConstantRankMatrix is a method function with options.