Next: 4.iii.b Quantum Schubert calculus

### 4.iii.a. General Schubert calculus

A flag in n-space is a sequence of linear subspaces F. : F1F2, ..., 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 a1a2, ..., 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