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4.iii.a. General Schubert calculus
A flag in n-space is a sequence of linear subspaces
F. : F1, F2, ..., Fn,
where Fi has dimension i and each subspace is
contained in the next.
Given a in Cn,k, the Schubert condition
of type a on a k-plane K imposed by the flag
F. is that
|
the intersection of K and
Fn+1-aj
has dimension at least k+1-j for each
j = 1, 2, ..., n .
|
(4.12) |
The Schubert subvariety Ya F. of
Gr(k,n) is the set of all
k-planes K satisfying the Schubert condition (4.12).
We relate this to the definitions of Section 4.ii.
Let e1, e2, ..., en
be a basis for Rn and for its
complexification Cn.
Defining Fi to be the linear span of
e1, e2, ..., ei gives
the standard flag F..
Then the Schubert variety Xav is
Ya F., where
avj = n+1-ak+1-j
for each j, and so
the codimension of Ya F.
is |a|.
A special Schubert condition is when
a = (1,...,k-1,k+l)
so that Ya F.
equals X(Fn-k+1-l ).
The general problem of the classical Schubert calculus of enumerative geometry
asks, given
a1, a2, ..., as
in Cn,k with
|a1|+|a2|+...+|as| = k(n-k) and general flags
F.1, F.2, ...,
F.s in Cn, how many
points are there in the intersection§ of the
Schubert varieties
|
Ya1 F.1,
Ya2 F.2,
...,
Yas F.s ?
|
(4.13) |
There are algorithms due to
Pieri [Pi] and
Giambelli [Gi] to
compute these numbers.
Other than the case when the indices ai are indices of
special Schubert varieties, it
remains open whether the general Schubert calculus is fully real.
(See [So7] and [So4] for some cases.)
§In this survey
flags are general when the corresponding intersection is
transverse..
Next: 4.iii.b Quantum Schubert calculus
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