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Schubert calculus for the Grassmannian Previous Next Contents

3.i. The Schubert calculus for the Grassmannian

Consider the following geometric problem. Given linear subspaces K1, K2, ..., Kn in m+p-space, how many p-planes H meet each Ki nontrivially? When the Ki are general, the numerical condition
mp = k1 + k2 + ... + kn
where m + 1 - ki = dim Ki 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 ,..., Fm+p with Fi contained in Fi+1 and dim Fi = 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 < ... < ap with ap 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 Fm+p+1-ai is at least i is called a Schubert variety (which we write as Ya F.), and this condition on p-planes H is called a Schubert condition. The codimension of the Schubert variety YaF. in the Grassmannian of p-planes in m+p-space is |a| := a1 - 1 + a2 - 2 + ... + ap - 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, So00]).


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