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Gromov-Witten invariants on Grassmannians


Authors: Anders Skovsted Buch, Andrew Kresch and Harry Tamvakis
Journal: J. Amer. Math. Soc. 16 (2003), 901-915
MSC (2000): Primary 14N35; Secondary 14M15, 14N15, 05E15
DOI: https://doi.org/10.1090/S0894-0347-03-00429-6
Published electronically: May 1, 2003
MathSciNet review: 1992829
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that any three-point genus zero Gromov-Witten invariant on a type $A$ Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type $A$, to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.


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

  • [BCM] L. Bégin, C. Cummins and P. Mathieu : Generating-function method for fusion rules, J. Math. Phys. 41 (2000), no. 11, 7640-7674. MR 2001m:17032
  • [BKMW] L. Bégin, A. N. Kirillov, P. Mathieu and M. A. Walton : Berenstein-Zelevinsky triangles, elementary couplings, and fusion rules, Lett. Math. Phys. 28 (1993), no. 4, 257-268. MR 94g:81159
  • [BMW] L. Bégin, P. Mathieu and M. A. Walton : $\widehat{{su}}(3)\sb k$ fusion coefficients, Modern Phys. Lett. A 7 (1992), no. 35, 3255-3265. MR 93j:81028
  • [BS] N. Bergeron and F. Sottile : Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), no. 2, 373-423. MR 2000d:05127
  • [Be] A. Bertram : Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289-305. MR 98j:14067
  • [BCF] A. Bertram, I. Ciocan-Fontanine and W. Fulton : Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728-746. MR 2000k:14042
  • [Bu1] A. S. Buch : Quantum cohomology of Grassmannians, Compositio Math., to appear.
  • [Bu2] A. S. Buch : A direct proof of the quantum version of Monk's formula, Proc. Amer. Math. Soc., to appear.
  • [BKT1] A. S. Buch, A. Kresch, and H. Tamvakis : Grassmannians, two-step flags, and puzzles, in preparation.
  • [BKT2] A. S. Buch, A. Kresch, and H. Tamvakis : Quantum Pieri rules for isotropic Grassmannians, in preparation.
  • [FK] S. Fomin and A. N. Kirillov : Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, pp. 147-182, Progr. Math. 172, Birkhäuser Boston, Boston, MA, 1999. MR 2001a:05152
  • [F] W. Fulton : Young tableaux, L.M.S. Student Texts 35, Cambridge Univ. Press, Cambridge, 1997. MR 99f:05119
  • [FP] W. Fulton and R. Pandharipande : Notes on stable maps and quantum cohomology, in: Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure. Math. 62, Part 2, Amer. Math. Soc., Providence (1997), pp. 45-96. MR 98m:14025
  • [HB] H. Hiller and B. Boe : Pieri formula for $SO_{2n+1}/U_n$ and $Sp_n/U_n$, Adv. Math. 62 (1986), 49-67. MR 87k:14058
  • [K] A. Knutson : Private communication.
  • [KTW] A. Knutson, T. Tao and C. Woodward : The honeycomb model of $GL(n)$ tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., to appear.
  • [KT1] A. Kresch and H. Tamvakis : Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom., to appear.
  • [KT2] A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, Compositio Math., to appear.
  • [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. MR 93h:05170
  • [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. MR 98g:14063
  • [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. MR 99d:14060
  • [S] F. Sottile : Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89-110. MR 97g:14035
  • [Tu] G. Tudose : On the combinatorics of $sl(n)$-fusion coefficients, preprint (2001).
  • [Y] A. Yong : Degree bounds in quantum Schubert calculus, Proc. Amer. Math. Soc., to appear.

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Additional Information

Anders Skovsted Buch
Affiliation: Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus C, Denmark
Email: abuch@imf.au.dk

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

Harry Tamvakis
Affiliation: Department of Mathematics, Brandeis University - MS 050, P. O. Box 9110, Waltham, Massachusetts 02454-9110
Email: harryt@brandeis.edu

DOI: https://doi.org/10.1090/S0894-0347-03-00429-6
Keywords: Gromov-Witten invariants, Grassmannians, Flag varieties, Schubert varieties, Quantum cohomology, Littlewood-Richardson rule
Received by editor(s): July 18, 2002
Published electronically: May 1, 2003
Additional Notes: The authors were supported in part by NSF Grant DMS-0070479 (Buch), an NSF Postdoctoral Research Fellowship (Kresch), and NSF Grant DMS-0296023 (Tamvakis).
Article copyright: © Copyright 2003 American Mathematical Society

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