Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
  Journal of Algebraic Geometry
Journal of Algebraic Geometry
  
Online ISSN 1534-7486; Print ISSN 1056-3911
 

Quantum cohomology of the Lagrangian Grassmannian


Authors: Andrew Kresch and Harry Tamvakis
Journal: J. Algebraic Geom. 12 (2003), 777-810
Published electronically: June 3, 2003
MathSciNet review: 1993764
Full-text PDF

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 [Enhancements On Off] (What's this?)

  • [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

DOI: http://dx.doi.org/10.1090/S1056-3911-03-00347-3
PII: S 1056-3911(03)00347-3
Received by editor(s): May 27, 2001
Published electronically: June 3, 2003


Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2014 University Press, Inc.
Comments: jag-query@ams.org
AMS Website