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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Degree bounds in quantum Schubert calculus


Author: Alexander Yong
Journal: Proc. Amer. Math. Soc. 131 (2003), 2649-2655
MSC (1991): Primary 14M15; Secondary 05E05, 14N10
Published electronically: January 8, 2003
MathSciNet review: 1974319
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Abstract | References | Similar Articles | Additional Information

Abstract: Fulton and Woodward have recently identified the smallest degree of $q$ that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result, and provide an alternative characterization of this smallest degree in terms of the rim hook formula for the quantum product.


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

  • 1. A. Bertram, Quantum Schubert Calculus, Adv. Math. 128 (1997) 289-305. MR 98j:14067
  • 2. A. Bertram, I. Ciocan-Fontanine and W. Fulton, Quantum Multiplication of Schur polynomials, Journal of Algebra 219 (1999) 728-746. MR 2000k:14042
  • 3. A. Buch, Quantum cohomology of Grassmannians, e-print math.AG/0106268.
  • 4. A. Buch, Littlewood-Richardson Calculator, software available at http://www-math. mit.edu/~abuch.
  • 5. W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209-249. MR 2001g:15023
  • 6. W. Fulton, Young tableaux, Cambridge University Press, 1997. MR 99f:05119
  • 7. W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry--Santa Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, 45-96. MR 98m:14025
  • 8. W. Fulton and C. Woodward, preprint, 2001.
  • 9. G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16. Addison-Wesley Co. Reading, Mass., 1981.
  • 10. M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Mirror Symmetry II, Amer. Math. Soc., Providence, RI, 1997, 607-653.
  • 11. I. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995. MR 96h:05207
  • 12. A. Postnikov, Affine approach to quantum Schubert calculus, e-print math. CO/0205165.
  • 13. Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994) 269-278.
  • 14. F. Sottile, Rational Curves on Grassmannians: systems theory, reality, and transverality, to appear in Contemporary Mathematics, 2001.
  • 15. R. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999. MR 2000k:05026

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Additional Information

Alexander Yong
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: ayong@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06850-3
PII: S 0002-9939(03)06850-3
Keywords: Gromov-Witten invariants, quantum cohomology, Grassmannian, Schubert calculus
Received by editor(s): December 14, 2001
Received by editor(s) in revised form: April 2, 2002
Published electronically: January 8, 2003
Communicated by: John R. Stembridge
Article copyright: © Copyright 2003 American Mathematical Society