15 June 2005 Affine approach to quantum Schubert calculus
Alexander Postnikov
Duke Math. J. 128(3): 473-509 (15 June 2005). DOI: 10.1215/S0012-7094-04-12832-5


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 S 3 , / n 2 , and / 2 . The last symmetry is a certain \emph{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.


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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


Published: 15 June 2005
First available in Project Euclid: 9 June 2005

zbMATH: 1081.14070
MathSciNet: MR2145741
Digital Object Identifier: 10.1215/S0012-7094-04-12832-5

Primary: 05E05
Secondary: 14M15 , 14N35

Rights: Copyright © 2005 Duke University Press


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Vol.128 • No. 3 • 15 June 2005
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