Abstract
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 , , and . The last symmetry is a certain \emph{curious duality} of the quantum cohomology which inverts the quantum parameter . Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.
Citation
Alexander Postnikov. "Affine approach to quantum Schubert calculus." Duke Math. J. 128 (3) 473 - 509, 15 June 2005. https://doi.org/10.1215/S0012-7094-04-12832-5
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