toSchubertBasis -- express cycles on a Grassmannian in terms of the Schubert basis

Synopsis

• Usage:

  toSchubertBasis c

• Inputs:
  • c, a ring element, An element of the intersection ring of a Grassmannian of $k$-dimensional subspaces of a rank-$n$ vector bundle
• Outputs:
  • An element $c'$ of a polynomial ring $B[s_\lambda]$ where $B$ is the base ring of G and $\lambda$ runs over all diagrams in a $k\times n$ rectangle (this is the "Schubert ring" of G, see schubertRing). The element $c'$ is the representation of $c$ in terms of the Schubert basis of the intersection ring of G over B.

Description

i1 : A = flagBundle({3,3},VariableNames => H)

o1 = A

o1 : a flag bundle with subquotient ranks {2:3}

i2 : S = first bundles A

o2 = S

o2 : an abstract sheaf of rank 3 on A

i3 : G = flagBundle({1,2},S,VariableNames => K)

o3 = G

o3 : a flag bundle with subquotient ranks {1..2}

i4 : RG = intersectionRing G

o4 = RG

o4 : QuotientRing

i5 : c = H_(2,3)*((K_(2,1))^2) + H_(1,1)*K_(2,2)

                                              2
o5 = - H   K    + (H   K    - H   H   K    - H   H    + H   H   )
        2,1 2,2     2,3 2,2    2,1 2,3 2,1    2,1 2,3    2,2 2,3

o5 : RG

i6 : toSchubertBasis c

         2
o6 = (- H   H    + H   H   )s    - H   H   s    + (H    - H   )s
         2,1 2,3    2,2 2,3  {0}    2,1 2,3 {1}     2,3    2,1  {2}

o6 : Schubert Basis of G(1,3) over A

See also

• schubertRing -- get the Schubert ring of a Grassmannian