Quantum cohomology of the Lagrangian Grassmannian
Authors:
Andrew Kresch and Harry Tamvakis
Journal:
J. Algebraic Geom. 12 (2003), 777-810
DOI:
https://doi.org/10.1090/S1056-3911-03-00347-3
Published electronically:
June 3, 2003
MathSciNet review:
1993764
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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.
[AS]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]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]Ber A. Bertram : Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305.
[BCF]BCF A. Bertram, I. Ciocan-Fontanine and W. Fulton : Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728–746.
[BH]HB 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]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]C L. Chen : Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), no. 1, 1–34.
[C-F1]CF1 I. Ciocan-Fontanine : The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2695–2729.
[C-F2]CF2 I. Ciocan-Fontanine : On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485–524.
[D1]D1 M. Demazure : Invariants symétriques des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301.
[D2]D2 M. Demazure : Désingularisation des variétés de Schubert généralisées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1974), 53–88.
[FGP]FGP S. Fomin, S. Gelfand and A. Postnikov : Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565–596.
[F]F W. Fulton : Intersection Theory, Second edition, Ergebnisse der Math. 2, Springer-Verlag, Berlin, 1998.
[FP]FPa 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]FP W. Fulton and P. Pragacz : Schubert varieties and degeneracy loci, Lecture Notes in Math. 1689, Springer-Verlag, Berlin, 1998.
[GK]GK A. Givental and B. Kim : Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609–641.
[G1]G 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]Groth 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]hartshorne R. Hartshorne : Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York, 1977.
[KL]KL G. Kempf and D. Laksov : The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162.
[K1]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]K2 B. Kim : On equivariant quantum cohomology, Internat. Math. Res. Notices 1996, no. 17, 841–851.
[K3]K3 B. Kim : Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), 129–148.
[KP]kimpandharipande 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]KM M. Kontsevich, Y. Manin : Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562.
[KT1]KT 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]KTorth A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, Compositio Math., to appear.
[LLT]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]LT J. Li and G. Tian : The quantum cohomology of homogeneous varieties, J. Algebraic Geom. 6 (1997), 269–305.
[LP]LP1 A. Lascoux and P. Pragacz : Operator calculus for $\tilde {Q}$-polynomials and Schubert polynomials, Adv. Math. 140 (1998), no. 1, 1–43.
[M]M I. G. Macdonald : Symmetric Functions and Hall Polynomials, Second edition, Clarendon Press, Oxford, 1995.
[P]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]PR P. Pragacz and J. Ratajski : Formulas for Lagrangian and orthogonal degeneracy loci; $\tilde {Q}$-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87.
[R]R J. Riordan : Combinatorial Identities, John Wiley & Sons, New York, 1968.
[S]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]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]St J. R. Stembridge : Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87–134.
[T]T2 H. Tamvakis : Arakelov theory of the Lagrangian Grassmannian, J. Reine Angew. Math. 516 (1999), 207-223.
[Th]thomsen J. Thomsen : Irreducibility of $\overline {M}_{0,n}(G/P,\beta )$, Internat. J. Math. 9, no. 3 (1998), 367–376.
[V]V C. Vafa : Topological mirrors and quantum rings, Essays on mirror manifolds, 96–119, Internat. Press, Hong Kong, 1992.
[W]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.
[AS]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]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]Ber A. Bertram : Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305.
[BCF]BCF A. Bertram, I. Ciocan-Fontanine and W. Fulton : Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728–746.
[BH]HB 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]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]C L. Chen : Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), no. 1, 1–34.
[C-F1]CF1 I. Ciocan-Fontanine : The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2695–2729.
[C-F2]CF2 I. Ciocan-Fontanine : On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485–524.
[D1]D1 M. Demazure : Invariants symétriques des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301.
[D2]D2 M. Demazure : Désingularisation des variétés de Schubert généralisées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1974), 53–88.
[FGP]FGP S. Fomin, S. Gelfand and A. Postnikov : Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565–596.
[F]F W. Fulton : Intersection Theory, Second edition, Ergebnisse der Math. 2, Springer-Verlag, Berlin, 1998.
[FP]FPa 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]FP W. Fulton and P. Pragacz : Schubert varieties and degeneracy loci, Lecture Notes in Math. 1689, Springer-Verlag, Berlin, 1998.
[GK]GK A. Givental and B. Kim : Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609–641.
[G1]G 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]Groth 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]hartshorne R. Hartshorne : Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York, 1977.
[KL]KL G. Kempf and D. Laksov : The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162.
[K1]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]K2 B. Kim : On equivariant quantum cohomology, Internat. Math. Res. Notices 1996, no. 17, 841–851.
[K3]K3 B. Kim : Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), 129–148.
[KP]kimpandharipande 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]KM M. Kontsevich, Y. Manin : Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562.
[KT1]KT 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]KTorth A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, Compositio Math., to appear.
[LLT]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]LT J. Li and G. Tian : The quantum cohomology of homogeneous varieties, J. Algebraic Geom. 6 (1997), 269–305.
[LP]LP1 A. Lascoux and P. Pragacz : Operator calculus for $\tilde {Q}$-polynomials and Schubert polynomials, Adv. Math. 140 (1998), no. 1, 1–43.
[M]M I. G. Macdonald : Symmetric Functions and Hall Polynomials, Second edition, Clarendon Press, Oxford, 1995.
[P]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]PR P. Pragacz and J. Ratajski : Formulas for Lagrangian and orthogonal degeneracy loci; $\tilde {Q}$-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87.
[R]R J. Riordan : Combinatorial Identities, John Wiley & Sons, New York, 1968.
[S]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]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]St J. R. Stembridge : Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87–134.
[T]T2 H. Tamvakis : Arakelov theory of the Lagrangian Grassmannian, J. Reine Angew. Math. 516 (1999), 207-223.
[Th]thomsen J. Thomsen : Irreducibility of $\overline {M}_{0,n}(G/P,\beta )$, Internat. J. Math. 9, no. 3 (1998), 367–376.
[V]V C. Vafa : Topological mirrors and quantum rings, Essays on mirror manifolds, 96–119, Internat. Press, Hong Kong, 1992.
[W]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
MR Author ID:
644754
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
Received by editor(s):
May 27, 2001
Published electronically:
June 3, 2003
