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Gromov-Witten invariants on Grassmannians

Authors: Anders Skovsted Buch, Andrew Kresch and Harry Tamvakis
Journal: J. Amer. Math. Soc. 16 (2003), 901-915
MSC (2000): Primary 14N35; Secondary 14M15, 14N15, 05E15
Published electronically: May 1, 2003
MathSciNet review: 1992829
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Abstract: We prove that any three-point genus zero Gromov-Witten invariant on a type $A$ Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type $A$, to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.

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Additional Information

Anders Skovsted Buch
Affiliation: Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus C, Denmark

Andrew Kresch
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395

Harry Tamvakis
Affiliation: Department of Mathematics, Brandeis University - MS 050, P. O. Box 9110, Waltham, Massachusetts 02454-9110

Keywords: Gromov-Witten invariants, Grassmannians, Flag varieties, Schubert varieties, Quantum cohomology, Littlewood-Richardson rule
Received by editor(s): July 18, 2002
Published electronically: May 1, 2003
Additional Notes: The authors were supported in part by NSF Grant DMS-0070479 (Buch), an NSF Postdoctoral Research Fellowship (Kresch), and NSF Grant DMS-0296023 (Tamvakis).
Article copyright: © Copyright 2003 American Mathematical Society