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Quantum cohomology of the Lagrangian Grassmannian

Author(s): Andrew Kresch; Harry Tamvakis
Journal: J. Algebraic Geom. 12 (2003), 777-810.
Posted: June 3, 2003
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Abstract | References | Additional information

Abstract: Let $V$ be a symplectic vector space and $LG$ be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in $V$. We give a presentation for the (small) quantum cohomology ring $QH^*(LG)$ and show that its multiplicative structure is determined by the ring of $\widetilde{Q}$-polynomials. We formulate a `quantum Schubert calculus' which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing the structure constants appearing in the quantum product of Schubert classes.

References:

[AS]
A. Astashkevich and V. Sadov : Quantum cohomology of partial flag manifolds $F_{n_1,\ldots,n_k}$, Comm. Math. Phys. 170 (1995), no. 3, 503-528.

[BGG]
I. N. Bernstein, I. M. Gelfand and S. I. Gelfand : Schubert cells and cohomology of the spaces $G/P$, Russian Math. Surveys 28 (1973), no. 3, 1-26.

[Be]
A. Bertram : Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289-305.

[BCF]
A. Bertram, I. Ciocan-Fontanine and W. Fulton : Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728-746.

[BH]
B. Boe and H. Hiller : Pieri formula for $SO_{2n+1}/U_n$ and $Sp_n/U_n$, Adv. Math. 62 (1986), 49-67.

[Bo]
A. Borel : Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207.

[C]
L. Chen : Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), no. 1, 1-34.

[C-F1]
I. Ciocan-Fontanine : The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2695-2729.

[C-F2]
I. Ciocan-Fontanine : On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485-524.

[D1]
M. Demazure : Invariants symétriques des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301.

[D2]
M. Demazure : Désingularisation des variétés de Schubert généralisées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1974), 53-88.

[FGP]
S. Fomin, S. Gelfand and A. Postnikov : Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565-596.

[F]
W. Fulton : Intersection Theory, Second edition, Ergebnisse der Math. 2, Springer-Verlag, Berlin, 1998.

[FP]
W. Fulton and R. Pandharipande : Notes on stable maps and quantum cohomology, in Algebraic Geometry (Santa Cruz, 1995), 45-96, Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997.

[FPr]
W. Fulton and P. Pragacz : Schubert varieties and degeneracy loci, Lecture Notes in Math. 1689, Springer-Verlag, Berlin, 1998.

[GK]
A. Givental and B. Kim : Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609-641.

[G1]
A. Grothendieck : Techniques de construction et théorèmes d'existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki 13 (1960/61), no. 221.

[G2]
A. Grothendieck : Techniques de construction en géométrie analytique V: Fibrés vectoriels, fibrés projectifs, fibrés en drapeaux, in Familles d'espaces complexes et fondements de la géométrie analytique, Séminaire Henri Cartan 13 (1960/61), exposé 12.

[H]
R. Hartshorne : Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York, 1977.

[KL]
G. Kempf and D. Laksov : The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153-162.

[K1]
B. Kim : Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices 1995, no. 1, 1-15.

[K2]
B. Kim : On equivariant quantum cohomology, Internat. Math. Res. Notices 1996, no. 17, 841-851.

[K3]
B. Kim : Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), 129-148.

[KP]
B. Kim and R. Pandharipande : The connectedness of the moduli space of maps to homogeneous spaces, in Symplectic geometry and mirror symmetry (Seoul, 2000), 187-201, World Sci. Publ., River Edge, NJ, 2001.

[KM]
M. Kontsevich, Y. Manin : Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562.

[KT1]
A. Kresch and H. Tamvakis : Double Schubert polynomials and degeneracy loci for the classical groups, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 6, 1681-1727.

[KT2]
A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, Compositio Math., to appear.

[LLT]
A. Lascoux, B. Leclerc and J.-Y. Thibon : Fonctions de Hall-Littlewood et polynômes de Kostka-Foulkes aux racines de l'unité, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 1-6.

[LT]
J. Li and G. Tian : The quantum cohomology of homogeneous varieties, J. Algebraic Geom. 6 (1997), 269-305.

[LP]
A. Lascoux and P. Pragacz : Operator calculus for $\widetilde{Q}$-polynomials and Schubert polynomials, Adv. Math. 140 (1998), no. 1, 1-43.

[M]
I. G. Macdonald : Symmetric Functions and Hall Polynomials, Second edition, Clarendon Press, Oxford, 1995.

[P]
P. Pragacz : Algebro-geometric applications of Schur $S$- and $Q$-polynomials, Séminare d'Algèbre Dubreil-Malliavin 1989-1990, Lecture Notes in Math. 1478, 130-191, Springer-Verlag, Berlin, 1991.

[PR]
P. Pragacz and J. Ratajski : Formulas for Lagrangian and orthogonal degeneracy loci; $\widetilde{Q}$-polynomial approach, Compositio Math. 107 (1997), no. 1, 11-87.

[R]
J. Riordan : Combinatorial Identities, John Wiley & Sons, New York, 1968.

[S]
I. Schur : Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250.

[ST]
B. Siebert and G. Tian : On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679-695.

[St]
J. R. Stembridge : Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87-134.

[T]
H. Tamvakis : Arakelov theory of the Lagrangian Grassmannian, J. Reine Angew. Math. 516 (1999), 207-223.

[Th]
J. Thomsen : Irreducibility of $\overline{M}_{0,n}(G/P,\beta)$, Internat. J. Math. 9, no. 3 (1998), 367-376.

[V]
C. Vafa : Topological mirrors and quantum rings, Essays on mirror manifolds, 96-119, Internat. Press, Hong Kong, 1992.

[W]
E. Witten : The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, & physics, 357-422, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995.

Additional Information:

Andrew Kresch
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd St., Philadelphia, Pennsylvania 19104-6395
Email: kresch@math.upenn.edu

Harry Tamvakis
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd St., Philadelphia, Pennsylvania 19104-6395
Address at time of publication: Department of Mathematics, Brandeis University, P. O. Box 9110, MS 050, Waltham, Massachusetts 02454-9110
Email: harryt@math.upenn.edu, harryt@brandeis.edu

PII: S 1056-3911(03)00347-3
Received by editor(s): May 27, 2001
Posted: June 3, 2003

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