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.

我们证明如果 $X$ 是 A 型的格拉斯曼，那么（小）量子上同调环的舒伯特基 $清华\left(X\right)$ 是普通上同调环的舒伯特基的唯一均匀变形 $H^{＊}\left(X\right)$乘以非负结构常数。这意味着（三点，属零）Gromov-Witten 不变量$X$ 由 Witten 的介绍唯一确定 $清华\left(X\right)$以及它们是非负的事实。我们推测任何旗帜品种都是如此$X=G∕磷$简单的花边谎言类型。对于各种完整的标志${ℂ}^n$，该猜想等价于 Fomin、Gelfand 和 Postnikov 的猜想，即 A 型量子舒伯特多项式由正性质唯一确定。我们对 Grassmannians 的证明回答了 Fulton 的问题。