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4.iii.a. General Schubert calculus
A flag in nspace is a sequence of linear subspaces
F. : F_{1}, F_{2}, ..., F_{n},
where F_{i} has dimension i and each subspace is
contained in the next.
Given a in C_{n,k}, the Schubert condition
of type a on a kplane K imposed by the flag
F. is that
the intersection of K and
F_{n+1aj}
has dimension at least k+1j for each
j = 1, 2, ..., n .

(4.12) 
The Schubert subvariety Y_{a} F. of
Gr(k,n) is the set of all
kplanes K satisfying the Schubert condition (4.12).
We relate this to the definitions of Section 4.ii.
Let e_{1}, e_{2}, ..., e_{n}
be a basis for R^{n} and for its
complexification C^{n}.
Defining F_{i} to be the linear span of
e_{1}, e_{2}, ..., e_{i} gives
the standard flag F..
Then the Schubert variety X_{av} is
Y_{a} F., where
a^{v}_{j} = n+1a_{k+1j}
for each j, and so
the codimension of Y_{a} F.
is a.
A special Schubert condition is when
a = (1,...,k1,k+l)
so that Y_{a} F.
equals X(F_{nk+1l} ).
The general problem of the classical Schubert calculus of enumerative geometry
asks, given
a^{1}, a^{2}, ..., a^{s}
in C_{n,k} with
a^{1}+a^{2}+...+a^{s} = k(nk) and general flags
F.^{1}, F.^{2}, ...,
F.^{s} in C^{n}, how many
points are there in the intersection^{§} of the
Schubert varieties
Y_{a1} F.^{1},
Y_{a2} F.^{2},
...,
Y_{as} 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 a^{i} 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
Up: 4.iii Further extensions of the Schubert
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