Positivity determines the quantum cohomology of Grassmannians

Buch, Anders Skovsted; Wang, Chengxi

doi:10.2140/ant.2021.15.1505

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Positivity determines the quantum cohomology of Grassmannians

Anders Skovsted Buch and Chengxi Wang

Vol. 15 (2021), No. 6, 1505–1521

DOI: 10.2140/ant.2021.15.1505

Abstract

We prove that if $X$ is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring $QH (X)$ is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring $H^{*} (X)$ that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov–Witten invariants of $X$ are uniquely determined by Witten’s presentation of $QH (X)$ and the fact that they are nonnegative. We conjecture that the same is true for any flag variety $X = G ∕ P$ of simply laced Lie type. For the variety of complete flags in $ℂ^{n}$ , this conjecture is equivalent to Fomin, Gelfand, and Postnikov’s conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton.

Keywords

quantum cohomology, Grassmannians, positivity, Gromov–Witten invariant, Schubert basis, quantum Schubert polynomials, flag varieties, symmetric functions, Seidel representation

Mathematical Subject Classification 2010

Primary: 14N35

Secondary: 05E05, 14M15, 14N15

Milestones

Received: 8 February 2020

Revised: 7 December 2020

Accepted: 5 January 2021

Published: 16 October 2021

Authors

Anders Skovsted Buch
Department of Mathematics Rutgers University Hill Center-Busch Campus Piscataway, NJ United States

Chengxi Wang
Mathematics Department Rutgers University Hill Center-Busch Campus Piscataway, NJ United States

Vol. 15, No. 6, 2021