schubertCycle' -- compute Schubert Cycles on a Grassmannian, Grothendieck-style

Synopsis

• Usage:

  schubertCycle'(s,F)

  F_s

• Inputs:
  • F, an abstract flag bundle, associated to a vector bundle $A$, say, of rank $n$. This flag bundle should be a Grassmannian, parametrizing rank $q$ quotient bundles of $A$, say.
  • s, a sequence $(s_0, ..., s_{q-1})$ of length $q$ of integers, with $0 \le{} s_0 < ... < s_{q-1} < n$, or a list $\{s_0, ..., s_{q-1}\}$ of length $q$ of integers, with $n-q \ge{} s_0 \ge{} ... \ge{} s_{q-1} \ge{} 0$
• Outputs:
  • c, the appropriate Schubert class or Schubert variety, depending on the the type of s. See page 271 of Fulton's book, Intersection Theory for the notation for schubertCycle, of which this is the dual. In the case where s is a a sequence, the value returned is the homology class of the Schubert variety in $ F $ consisting of those points corresponding to $(q-1)$-planes of $\PP(A)$ that meet $ W_j $ in dimension at least $ i $ with $0 \le{} i < q$, for each $ i $, where $ j = s_i $, and where $ W_j $ is the projective subspace of dimension $ j $ in a fixed (complete) flag $0 = W_0 \subset{} W_1 \subset{} ... \subset{} W_{n-1} = \PP(A) $. Here $ \PP(A) $ denotes the modern Grothendieck-style projective bundle parametrizing rank 1 quotient bundles of $A$; for the older "Fulton-style" version, see schubertCycle. In the case where s is a a list, the result is the corresponding Schubert class in cohomology. In Schubert2 homology and cohomology are identified with each other. Given a sequence $(s_0, ..., s_{q-1})$, one can convert it to a list yielding the same Schubert class by the formula $ \{..., n-q+i-s_i, ...\} $. This is related to schubertCycle as follows: if $E'$ is the dual bundle of $A$ and $G' = G(q,A')$ is the dual Grassmannian, then schubertCycle'(s,G) is carried to schubertCycle(s,G') under the duality isomorphism.

Description

i1 : base(0, Bundle=>(A, n=8, a))

o1 = a variety

o1 : an abstract variety of dimension 0

i2 : F = flagBundle ({5,q=3},A)

o2 = F

o2 : a flag bundle with subquotient ranks {5, 3}

i3 : CH = intersectionRing F;

i4 : F_(1,3,5)

              2      3
o4 = H   H   H    - H
      2,1 2,2 2,3    2,3

o4 : CH

i5 : {n-q+0-1,n-q+1-3,n-q+2-5}

o5 = {4, 3, 2}

o5 : List

i6 : F_oo

              2      3
o6 = H   H   H    - H
      2,1 2,2 2,3    2,3

o6 : CH