2019/05/14 by Anders Skovsted Buch, Chengxi Wang, Buch, Anders S. +1
Mathematics · #05E05 #14N15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Primary 14N35 #Secondary 14M15
paper · pdf · doi:10.48550/arxiv.1905.05723
openalex publication_date 2019/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 of X that multiplies with non-negative 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 non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, 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.