|
Quantum Schubert polynomials
Author(s):
Sergey
Fomin;
Sergei
Gelfand;
Alexander
Postnikov
Journal:
J. Amer. Math. Soc.
10
(1997),
565-596.
MSC (1991):
Primary 14M15;
Secondary 05E15, 14N10.
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
References |
Similar articles |
Additional information
References:
- 1.
- A. Astashkevich and V. Sadov, Quantum cohomology of partial flag manifolds
, Comm. Math. Phys. 170 (1995), 503-528. MR 96g:58027 - 2.
- I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Schubert cells and cohomology of the space
, Russian Math. Surveys 28 (1973), 1-26. MR 55:2941 - 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.
- A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207. MR 14:490e
- 6.
- C. Chevalley, Sur les décompositions cellulaires des espaces
, in: Algebraic Groups and Their Generalizations (W. Haboush and B. Parshall, eds.), Proc. Symp. Pure Math., vol. 56, Part 1, Amer. Math. Soc., Providence, RI, 1994, pp. 1-23. MR 95e:14041 - 7.
- I. Ciocan-Fontanine, Quantum cohomology of flag varieties, Intern. Math. Research Notices (1995), no. 6, 263-277. MR 96h:14071
- 8.
- M. Demazure, Désingularization des variétés de Schubert généralisées, Ann. Scient. Ecole Normale Sup. (4) 7 (1974), 53-88. MR 50:7174
- 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
-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.
- A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), 609-641. MR 96c:58027
- 16.
- B. Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairing, Intern. Math. Research Notices (1995), no. 1, 1-16. MR 96c:58028
- 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
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), 525-562. MR 95i:14049
- 20.
- B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight
, Selecta Math. (N.S.) 2 (1996), 43-91. CMP 96:16 - 21.
- A. Lascoux, Classes de Chern des varietes de drapeaux, C. R. Acad. Sci. Paris 295 (1982), 393-398. MR 85e:14074
- 22.
- A. Lascoux and M. P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris 294 (1982), 447-450. MR 83e:14039
- 23.
- A. Lascoux and M. P. Schützenberger, Fonctorialité de polynômes de Schubert, Contemp. Math. 88 (1989), 585-598. MR 90i:16003
- 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. 9 (1959), 253-286. MR 21:5641
- 27.
- P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; the
-polynomial approach, Compositio Math. (to appear). - 28.
- Y. Ruan and G. Tian, Mathematical theory of quantum cohomology, J. Diff. Geom. 42 (1995), no. 2, 259-367. MR 96m:58033
- 29.
- B. Sturmfels, Algorithms in invariant theory, Springer-Verlag, Berlin, 1993. MR 94m:13004
- 30.
- C. Vafa, Topological mirrors and quantum rings, in: Essays on Mirror Manifolds (S.-T. Yau, ed.), International Press, Boston, 1992. MR 94c:81193
- 31.
- E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry, vol. 1, International Press, Boston, 1991, pp. 243-310. MR 93e:32028
Similar Articles:
Retrieve articles in Journal of the American Mathematical Society
with MSC
(1991):
14M15,
05E15, 14N10.
Retrieve articles in all Journals with MSC
(1991):
14M15,
05E15, 14N10.
Additional Information:
Sergey
Fomin
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
fomin@math.mit.edu
Sergei
Gelfand
Affiliation:
American Mathematical Society, P.O.Box 6248, Providence, Rhode Island 02940-6248
Email:
sxg@ams.org
Alexander
Postnikov
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
apost@math.mit.edu
DOI:
10.1090/S0894-0347-97-00237-3
PII:
S 0894-0347(97)00237-3
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.
Copyright of article:
Copyright
1997,
American Mathematical Society
|