  This article 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 that the Gromov-Witten invariants of
the Grassmannian are invariant with respect to the action of a twisted product
of the groups S_3, (Z/nZ)^2, and Z/2Z. The last group gives a certain strange
duality of the quantum cohomologythat 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 that occur with
nonzero coefficients in the quantum product of two Schubert classes. The
strange 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.
