A Characterization of the Quantum Cohomology Ring of $G/B$ and Applications

http://dx.doi.org/10.4153/CJM-2008-037-8
Canad. J. Math. 60(2008), 875-891

  Published:2008-08-01
 Printed: Aug 2008

We observe that the small quantum product of the generalized flag manifold $G/B$ is a product operation $\star$ on $H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the facts that: it is a deformation of the cup product on $H^*(G/B)$; it is commutative, associative, and graded with respect to $\deg(q_i)=4$; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring $(H^*(G/B)\otimes \bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for $G/B$: the quantum Chevalley formula of D. Peterson (see also Fulton and Woodward ) and the ``quantization by standard monomials" formula of Fomin, Gelfand, and Postnikov for $G=\SL(n,\bC)$. The main idea of the proofs is the same as in Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can decode all information about the quantum cohomology of this space.