4.iii.c. Schubert calculus of flags

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 {a₁, a₂, ..., 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 w₁, w₂, ..., w_s in I_a with |w₁|+|w₂|+...+|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 w₁, w₂, ..., 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

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

(4.15)

Theorem 4.10 ([So8, Corollary 2.2]) Given a list I₁, I₂, ..., 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<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
Up: 4.iii Further extensions of the Schubert calculus: Table of Contents
Previous: 4.iii.b. Quantum Schubert calculus

---