Affine Approach to Quantum Schubert Calculus

This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the Gromov-Witten invariants of the Grassmannian with respect to the groups S3, (Z/nZ)2, and Z/2Z. The last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q . Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of q with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.