#### 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 $QH\left(X\right)$ is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring ${H}^{\ast }\left(X\right)$ 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 $QH\left(X\right)$ 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