Duke Mathematical Journal

Affine approach to quantum Schubert calculus

Alexander Postnikov

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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 $S_3$, $(\mathbb{Z}/n\mathbb{Z})^2$, and $\mathbb{Z}/2\mathbb{Z}$. 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.

Article information

Source
Duke Math. J. Volume 128, Number 3 (2005), 473-509.

Dates
First available: 9 June 2005

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1118341230

Digital Object Identifier
doi:10.1215/S0012-7094-04-12832-5

Mathematical Reviews number (MathSciNet)
MR2145741

Zentralblatt MATH identifier
1081.14070

Subjects
Primary: 05E05: Symmetric functions and generalizations
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Citation

Postnikov, Alexander. Affine approach to quantum Schubert calculus. Duke Mathematical Journal 128 (2005), no. 3, 473--509. doi:10.1215/S0012-7094-04-12832-5. http://projecteuclid.org/euclid.dmj/1118341230.


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