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

Mare, Augustin-Liviu

Canadian journal of mathematics, Tome 60 (2008) no. 4, pp. 875-891

Résumé

We observe that the small quantum product of the generalized flag manifold $G/B$ is a product operation $\star $ on ${{H}^{*}}(G/B)\otimes \mathbb{R}\left[ {{q}_{1,...,}}{{q}_{l}} \right]$ 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 $({{H}^{*}}(G/B)\otimes \mathbb{R}\left[ {{q}_{1,...,}}{{q}_{l}} \right]\star )$ 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 $G=SL(n,\mathbb{C})$ . The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum $\mathcal{D}$ -module of $G/B$ one can decode all information about the quantum cohomology of this space.

Détail

DOI : 10.4153/CJM-2008-037-8

Mots-clés : 14M15, 14N35

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