Vol. 15, No. 6, 2021

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Positivity determines the quantum cohomology of Grassmannians

Anders Skovsted Buch and Chengxi Wang

Vol. 15 (2021), No. 6, 1505–1521
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 = GP of simply laced Lie type. For the variety of complete flags in 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