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Quantum Schubert polynomials. (English) Zbl 0912.14018
{Let \(Fl_n\) be the manifold of complete flags in the \(n\)-dimensional vector space \(\mathbb C^n\). Inspired from ideas from string theory, recently the concept of quantum cohomology ring \(QH^*(X,\mathbb Z)\) of a Kähler algebraic manifold \(X\) has been defined. Then \[ QH^*(Fl_n,\mathbb Z)\cong H^*(Fl_n,\mathbb Z) \otimes\mathbb Z[q_1,\dots,q_{n-1}], \] where \(H^*(X,\mathbb Z)\) is the usual cohomology ring of \(Fl_n\) and \(q_1,\dots,q_{n-1}\) are formal variables (deformation parameters). So, the additive structures of the two cohomology rings are essentially the same. The multiplicative structure of \(H^*(X,\mathbb Z)\) can be recuperated from the multiplicative structure of \(QH^*(X,\mathbb Z)\) by taking \(q_1=\cdots=q_{n-1}=0\). The structure constants for the quantum cohomology are the 3-point Gromov-Witten invariants of genus zero. Recently, Givental, Kim and Ciocan-Lafontaine found a canonical isomorphism \[ QH^*(X,\mathbb Z)\cong\mathbb Z[q_1,\dots,q_{n-1}][x_1,\dots,x_n]/I_n^q, \] where \(x_1,\dots,x_n\) are variables and \(I_n^q\) is a certain ideal which can be explicitly described. This isomorphism extends an old isomorphism of Borel for the ordinary cohomology ring. The next problem naturally arising in the theory of quantum cohomology of the flag manifolds is to find an algebraic/combinatorial method for computing the structure constants of quantum multiplication in the basis of Schubert classes (the Gromov-Witten invariants). The aim of the paper under review is to solve this problem completely.}

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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[1] Alexander Astashkevich and Vladimir Sadov, Quantum cohomology of partial flag manifolds \?_{\?\(_{1}\)\cdots\?_{\?}}, Comm. Math. Phys. 170 (1995), no. 3, 503 – 528. · Zbl 0865.14027
[2] I. N. Bernšteĭn, I. M. Gel\(^{\prime}\)fand, and S. I. Gel\(^{\prime}\)fand, Schubert cells, and the cohomology of the spaces \?/\?, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3 – 26 (Russian).
[3] A. Bertram, Quantum Schubert calculus, Advances in Math. (to appear). · Zbl 0945.14031
[4] S. C. Billey and M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), 443-482. CMP 95:05 · Zbl 0832.05098
[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. · Zbl 0052.40001
[6] C. Chevalley, Sur les décompositions cellulaires des espaces \?/\?, 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. · Zbl 0824.14042
[7] Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263 – 277. · Zbl 0847.14011
[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. · Zbl 0312.14009
[9] C. Ehresmann, Sur la topologie de certains espaces homogènes, Ann. of Math. 35 (1934), 396-443. · JFM 60.1223.05
[10] P. di Francesco and C. Itzykson, Quantum intersection rings, in: The Moduli Space of Curves , Progress in Mathematics, vol. 129, Birkhäuser, Boston, 1995, pp. 81-148. · Zbl 0868.14029
[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. · Zbl 0878.14034
[14] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, preprint alg-geom/9608011. · Zbl 0898.14018
[15] Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609 – 641. · Zbl 0828.55004
[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. · Zbl 0849.14019
[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. · Zbl 1054.14533
[19] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525 – 562. · Zbl 0853.14020
[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). · Zbl 0495.14032
[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). · Zbl 0495.14031
[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). · Zbl 0697.20003
[24] J. Li and G. Tian, The quantum cohomology of homogeneous varieties, J. Algebraic Geom. (to appear). · Zbl 0909.14012
[25] I. G. Macdonald, Notes on Schubert polynomials, Publications LACIM, Montréal, 1991. · Zbl 0784.05061
[26] D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253 – 286. · Zbl 0096.36201
[27] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; the \(Q\)-polynomial approach, Compositio Math. (to appear). · Zbl 0916.14026
[28] Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259 – 367. · Zbl 0860.58005
[29] Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. · Zbl 0802.13002
[30] Cumrun Vafa, Topological mirrors and quantum rings, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 96 – 119. · Zbl 0827.58073
[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. · Zbl 0757.53049
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