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A direct proof of the quantum version of Monk's formula


Author: Anders Skovsted Buch
Journal: Proc. Amer. Math. Soc. 131 (2003), 2037-2042
MSC (2000): Primary 14N35; Secondary 14M15
DOI: https://doi.org/10.1090/S0002-9939-03-06765-0
Published electronically: January 28, 2003
MathSciNet review: 1963747
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Abstract: We use classical Schubert calculus to give a direct geometric proof of the quantum version of Monk's formula.

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

  • 1. A. S. Buch, Quantum cohomology of Grassmannians, to appear in Compositio Math.
  • 2. -, Quantum cohomology of partial flag varieties, in preparation, 2001.
  • 3. I. Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 1995, 263-277. MR 96h:14071
  • 4. -, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), 485-524. MR 2000d:14058
  • 5. S. Fomin, Lecture notes on quantum cohomology of the flag manifold, Publ. Inst. Math. (Beograd) (N.S.) 66(80) (1999), 91-100, Geometric combinatorics (Kotor, 1998). MR 2001g:05106
  • 6. S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565-596. MR 98d:14063
  • 7. S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Birkhäuser Boston, Boston, MA, 1999, pp. 147-182. MR 2001a:05152
  • 8. W. Fulton, Young tableaux, Cambridge University Press, 1997. MR 99f:05119
  • 9. W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry--Santa Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, pp. 45-96. MR 98m:14025
  • 10. A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), 609-641. MR 96c:58027
  • 11. B. Kim, Quantum cohomology of flag manifolds ${G}/{B}$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), 129-148. MR 2001c:14081
  • 12. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Mirror symmetry, II, Amer. Math. Soc., Providence, RI, 1997, pp. 607-653.
  • 13. I. G. Macdonald, Notes on Schubert polynomials, Laboratoire de Combinatoire et d'Informatique Mathématique, Université du Québec à Montréal, 1991.
  • 14. D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253-286. MR 21:5641
  • 15. A. Postnikov, On a quantum version of Pieri's formula, Advances in geometry, Birkhäuser Boston, Boston, MA, 1999, pp. 371-383. MR 99m:14096
  • 16. Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994), 269-278. MR 95b:58025

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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

DOI: https://doi.org/10.1090/S0002-9939-03-06765-0
Received by editor(s): December 18, 2001
Received by editor(s) in revised form: February 22, 2002, and March 6, 2002
Published electronically: January 28, 2003
Additional Notes: The author was partially supported by NSF Grant DMS-0070479
Communicated by: John R. Stembridge
Article copyright: © Copyright 2003 American Mathematical Society

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