Duke Mathematical Journal

Quantum cohomology rings of Grassmannians and total positivity

Konstanze Rietsch

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Abstract

We give a proof of a result of D. Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of ${\rm GL}_n$. The totally positive part of this subvariety is then constructed, and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of C. Vafa, K. Intriligator, and A. Bertram for the structure constants (Gromov-Witten invariants). Finally, we use the positivity of these Gromov-Witten invariants to prove certain inequalities for Schur polynomials at roots of unity.

Article information

Source
Duke Math. J., Volume 110, Number 3 (2001), 523-553.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087574980

Digital Object Identifier
doi:10.1215/S0012-7094-01-11033-8

Mathematical Reviews number (MathSciNet)
MR1869115

Zentralblatt MATH identifier
1013.14014

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 20G20: Linear algebraic groups over the reals, the complexes, the quaternions

Citation

Rietsch, Konstanze. Quantum cohomology rings of Grassmannians and total positivity. Duke Math. J. 110 (2001), no. 3, 523--553. doi:10.1215/S0012-7094-01-11033-8. https://projecteuclid.org/euclid.dmj/1087574980


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