For $n=1$, this function returns the kernel of the matrix describing 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 $V=<v_0,v_1>$, while for $n>1$, the function returns the kernel of the matrix describing 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 $SL(n+1)$.
In the paper A construction of equivariant bundles on the space of symmetric forms, it is proved that the matrix $\Phi$ has constant co-rank 1, so that the kernel $W = ker \Phi$ turns out to be a vector bundle.
i1 : n = 1, d = 3, m = 3 o1 = (1, 3, 3) o1 : Sequence |
i2 : W = slEquivariantVectorBundle(n,d,m) o2 = cokernel {4} | 0 0 0 0 0 x_3 x_2 | {4} | x_1 x_0 0 0 0 0 0 | {4} | -2x_2 3x_1 x_0 0 0 0 0 | {4} | x_3 -9x_2 0 x_0 0 0 0 | {4} | 0 5x_3 -3x_2 -x_1 2x_0 0 0 | {4} | 0 0 2x_3 -x_2 -3x_1 5x_0 0 | {4} | 0 0 0 x_3 0 -9x_1 x_0 | {4} | 0 0 0 0 x_3 3x_2 -2x_1 | 8 o2 : coherent sheaf on Proj(QQ[x ..x ]), quotient of OO (-4) 0 3 Proj(QQ[x ..x ]) 0 3 |
By default, slEquivariantVectorBundle defines the vector bundle over a projective space whose coordinate ring has 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 : W = slEquivariantVectorBundle(n,d,m,CoefficientRing=>ZZ/10007) o4 = cokernel {4} | 0 0 0 0 0 556x_3 -5003x_2 | {4} | x_1 x_0 0 0 0 0 0 | {4} | -x_2 -5002x_1 x_0 0 0 0 0 | {4} | 1668x_3 5002x_2 0 x_0 0 0 0 | {4} | 0 -1667x_3 -x_2 -x_1 x_0 0 0 | {4} | 0 0 4448x_3 3335x_2 -x_1 x_0 0 | {4} | 0 0 0 371x_3 0 -x_1 x_0 | {4} | 0 0 0 0 -2409x_3 1668x_2 -x_1 | ZZ 8 o4 : coherent sheaf on Proj(-----[x ..x ]), quotient of OO (-4) 10007 0 3 ZZ Proj(-----[x ..x ]) 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 : W = slEquivariantVectorBundle(R,d,m) 1 o7 = OO (-2) Proj(QQ[x ..x ]) 0 2 o7 : coherent sheaf on Proj(QQ[x ..x ]), free 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 vector bundle is defined over the projective space Proj(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 : W = slEquivariantVectorBundle(n,d,m,X) o10 = cokernel {4} | 0 0 0 0 0 2z_3 -3z_2 | {4} | z_1 z_0 0 0 0 0 0 | {4} | -z_2 -2z_1 z_0 0 0 0 0 | {4} | -z_3 2z_2 0 z_0 0 0 0 | {4} | 0 2z_3 -z_2 -z_1 z_0 0 0 | {4} | 0 0 2z_3 -3z_2 -z_1 z_0 0 | {4} | 0 0 0 -3z_3 0 -z_1 z_0 | {4} | 0 0 0 0 z_3 -z_2 -z_1 | 8 o10 : coherent sheaf on Proj X, quotient of OO (-4) Proj X |
i11 : R = QQ[y_0,y_1]; |
i12 : d = 3, m = 2 o12 = (3, 2) o12 : Sequence |
i13 : W = slEquivariantVectorBundle(R,d,m,X) o13 = cokernel {2} | 0 -2z_3 0 z_2 | {2} | z_1 0 z_0 0 | {2} | -z_2 z_0 0 0 | {2} | -2z_3 -z_1 -z_2 z_0 | {2} | 0 0 -2z_3 -z_1 | 5 o13 : coherent sheaf on Proj X, quotient of OO (-2) Proj X |
The object slEquivariantVectorBundle is a method function with options.