Quantum Schubert polynomials

Fomin, Sergey; Gelfand, Sergei; Postnikov, Alexander

doi:10.1090/S0894-0347-97-00237-3

Quantum Schubert polynomials
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by Sergey Fomin, Sergei Gelfand and Alexander Postnikov

J. Amer. Math. Soc. 10 (1997), 565-596

DOI: https://doi.org/10.1090/S0894-0347-97-00237-3

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References

Alexander Astashkevich and Vladimir Sadov, Quantum cohomology of partial flag manifolds $F_{n_1\cdots n_k}$, Comm. Math. Phys. 170 (1995), no. 3, 503–528. MR 1337131, DOI 10.1007/BF02099147
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
A. Bertram, Quantum Schubert calculus, Advances in Math. (to appear).
S. C. Billey and M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), 443–482.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
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, DOI 10.1090/pspum/056.1/1278698
Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263–277. MR 1344348, DOI 10.1155/S1073792895000213
Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697, DOI 10.24033/asens.1261
C. Ehresmann, Sur la topologie de certains espaces homogènes, Ann. of Math. 35 (1934), 396–443.
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
S. Fomin and A. N. Kirillov, Combinatorial $B_n$-analogues of Schubert polynomials, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3591–3620.
S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, AMS electronic preprint AMSPPS #199605-14-008, April 1996.
W. Fulton, Young tableaux with applications to representation theory and geometry, Cambridge University Press, 1996.
W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, preprint alg-geom/9608011.
Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609–641. MR 1328256, DOI 10.1007/BF02101846
Bumsig Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices 1 (1995), 1–15. MR 1317639, DOI 10.1155/S1073792895000018
B. Kim, On equivariant quantum cohomology, Intern. Math. Research Notices (1996), no. 17, 841–851.
B. Kim, Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, preprint alg-geom/9607001.
M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244, DOI 10.1007/BF02101490
B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight $\rho$, Selecta Math. (N.S.) 2 (1996), 43–91.
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
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
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, DOI 10.1090/conm/088/1000001
J. Li and G. Tian, The quantum cohomology of homogeneous varieties, J. Algebraic Geom. (to appear).
I. G. Macdonald, Notes on Schubert polynomials, Publications LACIM, 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
P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; the $Q$-polynomial approach, Compositio Math. (to appear).
Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259–367. MR 1366548
Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR 1255980, DOI 10.1007/978-3-7091-4368-1
Cumrun Vafa, Topological mirrors and quantum rings, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 96–119. MR 1191421
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