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 |
The object slEquivariantConstantRankMatrix is a method function with options.