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Quantum cohomology of orthogonal Grassmannians. (English) Zbl 1077.14083

Let \(V\) be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in \(V\). The authors give a presentation for the small quantum cohomology ring \(QH^*(\text{OG})\) and show that its product structure is determined by the ring of \(\tilde P\)-polynomials of P. Pragacz and J. Ratajski [Compos. Math. 107, 11–87 (1997; Zbl 0916.14026)]. 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, it is shown 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.
In a companion paper to this one [A. Kresch and H. Tamvakis, J. Algebr. Geom., 12, No. 4, 777–810 (2003; Zbl 1051.53070)], the authors provide an analogous analysis for the Lagrangian Grassmannian. The situation in the orthogonal case is similar, but with significant differences, both in the results and in their proofs.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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