A direct proof of the quantum version of Monk’s formula
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- by Anders Skovsted Buch
- Proc. Amer. Math. Soc. 131 (2003), 2037-2042
- DOI: https://doi.org/10.1090/S0002-9939-03-06765-0
- Published electronically: January 28, 2003
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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
- A. S. Buch, Quantum cohomology of Grassmannians, to appear in Compositio Math.
- —, Quantum cohomology of partial flag varieties, in preparation, 2001.
- Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263–277. MR 1344348, DOI 10.1155/S1073792895000213
- Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485–524. MR 1695799, DOI 10.1215/S0012-7094-99-09815-0
- Sergey 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 1765041
- Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565–596. MR 1431829, DOI 10.1090/S0894-0347-97-00237-3
- Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182. MR 1667680
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI 10.1090/pspum/062.2/1492534
- Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609–641. MR 1328256
- Bumsig Kim, Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), no. 1, 129–148. MR 1680543, DOI 10.2307/121021
- 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.
- I. G. Macdonald, Notes on Schubert polynomials, Laboratoire de Combinatoire et d’Informatique Mathématique, Université du Québec à Montréal, 1991.
- D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253–286. MR 106911, DOI 10.1112/plms/s3-9.2.253
- Alexander Postnikov, On a quantum version of Pieri’s formula, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 371–383. MR 1667687
- Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994), no. 2, 269–278. MR 1266766, DOI 10.4310/MRL.1994.v1.n2.a15
Bibliographic Information
- Anders Skovsted Buch
- Affiliation: Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus, C, Denmark
- MR Author ID: 607314
- Email: abuch@imf.au.dk
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1963747