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4.iii.c. Schubert calculus of flags
Let a:=0<a_{1}<...<a_{r}<a_{r+1}=n be a sequence of integers.
The manifold of partial flags in nspace (or the flag manifold)
Fl_{a}
is the collection of partial flags of subspaces
E. : E_{a1},
E_{a2}, ...,
E_{ar}
in nspace, where E_{ai} has dimension
a_{i}.
A complete flag F. is when
a=1,2,...,n1,n.
The Schubert varieties of Fl_{a} are indexed by
permutations w in the symmetric group on n letters, whose
descents only occur at positions in a.
That is, w(i)>w(i+1) implies that
i is in {a_{1}, a_{2}, ...,
a_{r}}.
Let I_{a} be this set of indices.
For a complete flag F. and w in
I_{a}, the partial flag manifold
Fl_{a} has a Schubert variety
Y_{w} F. given by
all partial flags E. satisfying
The intersection of E_{aj} and
F_{i} is at least
#{ l < a_{j}+1  w(l) < i+1 } .
The codimension of Y_{w} F.
is w:=#{i<j  w(i)>w(j)}.
We state the general question in the Schubert calculus of enumerative geometry
for flags.
Question 4.9
Given permutations
w_{1},
w_{2}, ...,
w_{s} in
I_{a} with

w_{1}+
w_{2}+...+
w_{s}
equal to the dimension of the manifold of partial flags
Fl_{a}, and general flags
F.^{1},
F.^{2}, ...,
F.^{s}, what is the number of points in the
intersection of the Schubert varieties
Y_{w1} F.^{1},
Y_{w2} F.^{2},
...,
Y_{ws} F.^{s} ?
That is, how many partial flags satisfy the Schubert conditions
w_{1},
w_{2}, ...,
w_{s}
imposed the (respective) flags
F.^{1},
F.^{2}, ...,
F.^{s}.
There are algorithms [BGG,De] for computing this number and the
numbers for the orthogonal and the Lagrangian Schubert calculus in
Sections 4.iii.d
and 4.iii.e.
When w=(i,i+1),
Y_{w} F.
is the simple Schubert variety, written
Y_{i} F. and defined by
Y_{i} F. :=
{E.  E_{i} meets
F_{ni} nontrivially } .

(4.15) 
Theorem 4.10 ([
So8,
Corollary 2.2])
Given a list
I_{1},
I_{2}, ...,
I_{N} (
N = dim
Fl_{a}) of numbers
with
i_{j} in
a, there exist real flags
F.^{1},
F.^{2}, ...,
F.^{N} such that the intersection of the simple
Schubert varieties
Y_{i1} F.^{1},
Y_{i2} F.^{2},
...,
Y_{iN} F.^{N}
is transverse and consists only of real flags.
The case of a=2<n2 of this theorem was proven
earlier [So2, Theorem 13].
It remains open whether the general Schubert calculus of flags is fully real.
Next: 4.iii.d Orthogonal Schubert calculus
Up: 4.iii Further extensions of the Schubert
calculus: Table of Contents
Previous: 4.iii.b. Quantum Schubert calculus