A characterization of the quantum cohomology ring of G/B and applications

We show that the small quantum product of the generalized flag manifold $G\mathrm{/}B$ is a product operation on $H^{\ast }(G\mathrm{/}B)\otimes \text{bR}[q_1,\mathrm{.}\mathrm{.}\mathrm{.},q_l]$ uniquely determined by the fact that it is a deformation of the cup product on $H^{\ast }(G\mathrm{/}B)$, it is commutative, associative, graded with respect to $\mathrm{deg}(q_i)=4$, it satisfies a certain relation (of degree two), and the corresponding Dubrovin connection is flat. We deduce that it is again the flatness of the Dubrovin connection which characterizes essentially the solutions of the “quantum Giambelli problem” for $G\mathrm{/}B$. This result gives new proofs of the quantum Chevalley formula (proved by Peterson and Fulton-Woodward), and of Fomin, Gelfand and Postnikov’s description of the quantization map for $F{l}_n$.