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### 4.iii.c. Schubert calculus of flags

Let a:=0<a1<...<ar<ar+1=n be a sequence of integers. The manifold of partial flags in n-space (or the flag manifold) Fla is the collection of partial flags of subspaces

E. : Ea1, Ea2, ..., Ear

in n-space, where Eai has dimension ai. A complete flag F. is when a=1,2,...,n-1,n.

The Schubert varieties of Fla 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 {a1, a2, ..., ar}. Let Ia be this set of indices. For a complete flag F. and w in Ia, the partial flag manifold Fla has a Schubert variety Yw F. given by all partial flags E. satisfying

The intersection of Eaj and Fi is at least #{ l < aj+1 | w(l) < i+1 } .

The codimension of Yw 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 w1, w2, ..., ws in Ia with |w1|+|w2|+...+|ws| equal to the dimension of the manifold of partial flags Fla, and general flags F.1, F.2, ..., F.s, what is the number of points in the intersection of the Schubert varieties

Yw1 F.1,   Yw2 F.2,  ...,   Yws F.s ?

That is, how many partial flags satisfy the Schubert conditions w1, w2, ..., ws 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), Yw F. is the simple Schubert variety, written Yi F. and defined by

 Yi F.   :=   {E. | Ei meets Fn-i non-trivially } . (4.15)

Theorem 4.10 ([So8, Corollary 2.2]) Given a list I1, I2, ..., IN (N = dim Fla) of numbers with ij in a, there exist real flags F.1, F.2, ..., F.N such that the intersection of the simple Schubert varieties

Yi1 F.1, Yi2 F.2, ..., YiN F.N

is transverse and consists only of real flags.

The case of a=2<n-2 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