This function returns the kernel of the matrix describing the morphism
Φ: Smd-2V ⊗Oℙd →S(m-1)dV ⊗Oℙd)(1)
given by the projection
SdV ⊗S(m-1)dV →Smd-2V
of the irreducible SL(2)-subrepresentation of highest weight md-2, where ℙd = ℙ(SdV) as V=<v0,v1>.
In the paper A construction of equivariant bundles on the space of symmetric forms, it is proved that the matrix Φ has constant co-rank 1, so that the kernel W = ker Φ turns out to be a vector bundle, and the entries of the matrix Φ are explicitely describred.
i1 : d = 3, m = 2 o1 = (3, 2) o1 : Sequence |
i2 : W = sl2EquivariantVectorBundle(d,m) o2 = cokernel {4} | 0 x_3 0 x_2 | {4} | x_1 0 x_0 0 | {4} | -x_2 x_0 0 0 | {4} | x_3 -3x_1 -3x_2 x_0 | {4} | 0 0 x_3 -x_1 | 5 o2 : coherent sheaf on Proj(QQ[x , x , x , x ]), quotient of OO (-4) 0 1 2 3 Proj(QQ[x , x , x , x ]) 0 1 2 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 : d = 3, m = 2 o3 = (3, 2) o3 : Sequence |
i4 : W = sl2EquivariantVectorBundle(d,m,CoefficientRing=>ZZ/10007) o4 = cokernel {4} | 0 3336x_3 0 x_2 | {4} | x_1 0 x_0 0 | {4} | -x_2 x_0 0 0 | {4} | 3336x_3 -x_1 -x_2 x_0 | {4} | 0 0 3336x_3 -x_1 | ZZ 5 o4 : coherent sheaf on Proj(-----[x , x , x , x ]), quotient of OO (-4) 10007 0 1 2 3 ZZ Proj(-----[x , x , x , x ]) 10007 0 1 2 3 |
If the first argument is a polynomial ring R, then d = numgens R-1.
i5 : R = QQ[y_0..y_3]; |
i6 : m = 2 o6 = 2 |
i7 : W = sl2EquivariantVectorBundle(R,m) o7 = cokernel {4} | 0 y_3 0 y_2 | {4} | y_1 0 y_0 0 | {4} | -y_2 y_0 0 0 | {4} | y_3 -3y_1 -3y_2 y_0 | {4} | 0 0 y_3 -y_1 | 5 o7 : coherent sheaf on Proj(R), quotient of OO (-4) Proj(R) |