Abstract
We prove that if is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov–Witten invariants of are uniquely determined by Witten’s presentation of and the fact that they are nonnegative. We conjecture that the same is true for any flag variety of simply laced Lie type. For the variety of complete flags in , 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.
Citation
Anders Skovsted Buch. Chengxi Wang. "Positivity determines the quantum cohomology of Grassmannians." Algebra Number Theory 15 (6) 1505 - 1521, 2021. https://doi.org/10.2140/ant.2021.15.1505
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