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SLnEquivariantMatrices :: sl2EquivariantVectorBundle

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

Synopsis

Description

This function returns the kernel of the matrix describing the morphism

Φ: Smd-2V ⊗Od →S(m-1)dV ⊗Od)(1)

given by the projection

SdV ⊗S(m-1)dV →Smd-2V

of the irreducible SL(2)-subrepresentation of highest weight md-2, where d = 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

Ways to use sl2EquivariantVectorBundle :