Compositio Mathematica

Quantum cohomology of orthogonal Grassmannians

Andrew Kresch a1 and Harry Tamvakis a2
a1 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA
a2 Department of Mathematics, Brandeis University, MS 050, PO Box 9110, Waltham, MA 02454-9110, USA

Article author query
kresch a   [Google Scholar] 
tamvakis h   [Google Scholar] 


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.

(Received April 12 2002)
(Accepted March 3 2003)

Key Words: quantum cohomology; Quot schemes; Schubert calculus.

Maths Classification

14M15; 05E15.