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Quantum cohomology of orthogonal Grassmannians

Published online by Cambridge University Press:  04 December 2007

Andrew Kresch  and
Harry Tamvakis
Andrew Kresch
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, [email protected]
Harry Tamvakis
Affiliation:
Department of Mathematics, Brandeis University, MS 050, PO Box 9110, Waltham, MA 02454-9110, [email protected]
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Abstract

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Let V be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH*(OG) and show that its product structure is determined by the ring of $\widetilde{P}$-polynomials. A ‘quantum Schubert calculus’ is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov--Witten invariants. As an application, we show that the table of three-point, genus zero Gromov–Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.

Keywords

quantum cohomologyQuot schemesSchubert calculus14M1505E15
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 2 , March 2004 , pp. 482 - 500
Copyright
Foundation Compositio Mathematica 2004