sl2EquivariantVectorBundle -- computes a SL(2)-equivariant vector bundle over some projective space

Synopsis

Usage:

W = sl2EquivariantVectorBundle(d,m)

W = sl2EquivariantVectorBundle(d,m,CoefficientRing=>C)

W = sl2EquivariantVectorBundle(R,m)
Inputs:
- R, a polynomial ring, with at least two variables
- d, an integer, positive
- m, an integer, at least 2
- C, a ring,
Optional inputs:
- CoefficientRing => ..., default value QQ, name for optional argument
Outputs:
- W, a coherent sheaf, vector bundle

Description

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 $V=<v_0,v_1>$.

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, and the entries of the matrix $\Phi$ are explicitly described.

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 ]), 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 : 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 ]), quotient of OO                    (-4)
                            10007  0   3                         ZZ
                                                          Proj(-----[x ..x ])
                                                               10007  0   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

Ways to use sl2EquivariantVectorBundle:

sl2EquivariantVectorBundle(PolynomialRing,ZZ)
sl2EquivariantVectorBundle(ZZ,ZZ)

For the programmer

The object sl2EquivariantVectorBundle is a method function with options.