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Quantum Schubert polynomials

Authors: Sergey Fomin, Sergei Gelfand and Alexander Postnikov
Journal: J. Amer. Math. Soc. 10 (1997), 565-596
MSC (1991): Primary 14M15; Secondary 05E15, 14N10.
MathSciNet review: 1431829
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  • 1. Alexander Astashkevich and Vladimir Sadov, Quantum cohomology of partial flag manifolds 𝐹_{𝑛₁\cdots𝑛_{𝑘}}, Comm. Math. Phys. 170 (1995), no. 3, 503–528. MR 1337131
  • 2. I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces 𝐺/𝑃, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
  • 3. A. Bertram, Quantum Schubert calculus, Advances in Math. (to appear).
  • 4. S. C. Billey and M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), 443-482. CMP 95:05
  • 5. Armand 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 (French). MR 0051508
  • 6. C. Chevalley, Sur les décompositions cellulaires des espaces 𝐺/𝐵, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1–23 (French). With a foreword by Armand Borel. MR 1278698
  • 7. Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263–277. MR 1344348, 10.1155/S1073792895000213
  • 8. Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I. MR 0354697
  • 9. C. Ehresmann, Sur la topologie de certains espaces homogènes, Ann. of Math. 35 (1934), 396-443.
  • 10. P. di Francesco and C. Itzykson, Quantum intersection rings, in: The Moduli Space of Curves (R. Dijkgraaf, C. Faber, and G. van der Geer, eds.), Progress in Mathematics, vol. 129, Birkhäuser, Boston, 1995, pp. 81-148. MR 96k:14041
  • 11. S. Fomin and A. N. Kirillov, Combinatorial $B_n$-analogues of Schubert polynomials, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3591-3620. CMP 96:15
  • 12. S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, AMS electronic preprint AMSPPS #199605-14-008, April 1996.
  • 13. W. Fulton, Young tableaux with applications to representation theory and geometry, Cambridge University Press, 1996.
  • 14. W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, preprint alg-geom/9608011.
  • 15. Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609–641. MR 1328256
  • 16. Bumsig Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices 1 (1995), 1–15 (electronic). MR 1317639, 10.1155/S1073792895000018
  • 17. B. Kim, On equivariant quantum cohomology, Intern. Math. Research Notices (1996), no. 17, 841-851. CMP 97:04
  • 18. B. Kim, Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, preprint alg-geom/9607001.
  • 19. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
  • 20. B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight $\rho $, Selecta Math. (N.S.) 2 (1996), 43-91. CMP 96:16
  • 21. Alain Lascoux, Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 5, 393–398 (French, with English summary). MR 684734
  • 22. Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739, 10.1090/conm/088/1000001
  • 23. A. Lascoux and M.-P. Schützenberger, Fonctorialité des polynômes de Schubert, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 585–598 (French, with English summary). MR 1000001, 10.1090/conm/088/1000001
  • 24. J. Li and G. Tian, The quantum cohomology of homogeneous varieties, J. Algebraic Geom. (to appear).
  • 25. I. G. Macdonald, Notes on Schubert polynomials, Publications LACIM, Montréal, 1991.
  • 26. D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253–286. MR 0106911
  • 27. P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; the $Q$-polynomial approach, Compositio Math. (to appear).
  • 28. Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259–367. MR 1366548
  • 29. Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR 1255980
  • 30. Cumrun Vafa, Topological mirrors and quantum rings, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 96–119. MR 1191421
  • 31. Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR 1144529

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Additional Information

Sergey Fomin
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Sergei Gelfand
Affiliation: American Mathematical Society, P.O.Box 6248, Providence, Rhode Island 02940-6248

Alexander Postnikov
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Keywords: Gromov-Witten invariants, quantum cohomology, flag manifold, Schubert polynomials
Received by editor(s): July 8, 1996
Received by editor(s) in revised form: December 23, 1996
Additional Notes: The first author was supported in part by NSF grant DMS-9400914.
Article copyright: © Copyright 1997 American Mathematical Society