Schubert calculus for the Grassmannian

3.i. The Schubert calculus for the Grassmannian

Consider the following geometric problem. Given linear subspaces K₁, K₂, ..., K_n in m+p-space, how many p-planes H meet each K_i nontrivially? When the K_i are general, the numerical condition

mp = k₁ + k₂ + ... + k_n

where m + 1 - k_i = dim K_i must be satisfied in order for there to be finitely many. We call this a Pieri-type enumerative problem.

More generally, we seek p-planes having fixed position with respect to some general flags. Let F. be a flag, a sequence of subspaces F₁, F₂ ,..., F_m+p with F_i contained in F_i₊₁ and dim F_i = i. The possible positions of p-planes H with respect to a flag F. are encoded by increasing sequences of positive integers a : a₁ < a₂ < ... < a_p with a_p at most m+p.

Given such a sequence a and a flag F., the set of all p-planes H where the dimension of the intersection of H with F_{m+p+1-a_i} is at least i is called a Schubert variety (which we write as Y_a F.), and this condition on p-planes H is called a Schubert condition. The codimension of the Schubert variety Y_aF. in the Grassmannian of p-planes in m+p-space is |a| := a₁ - 1 + a₂ - 2 + ... + a_p - p.

Given sequences a¹, a², ..., aⁿ with |a¹| + |a²| + ... + |aⁿ| = mp and flags F.¹, F.², ..., F.ⁿ in general position, there are finitely many p-planes H satisfying the Schubert conditions with respect to the given flags. Such a collection of sequences are called Schubert data.

Schubert's calculus is a collection of algorithms for computing these intersection numbers. We will not describe how to do this here. For details, we refer to the literature ([HP, So98]).