The Quantum Cohomology of Flag Varieties and the Periodicity of the Schubert Structure Constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F (k1, . . . , kr; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. As a corollary to the main theorem in [C3], we obtain a positive, geometric rule for computing these invariants. Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.