Quantum double Schubert polynomials represent Schubert classes

Lam, Thomas; Shimozono, Mark

doi:10.1090/S0002-9939-2013-11831-9

Quantum double Schubert polynomials represent Schubert classes
HTML articles powered by AMS MathViewer

by Thomas Lam and Mark Shimozono PDF

Proc. Amer. Math. Soc. 142 (2014), 835-850 Request permission

Abstract:

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim’s presentation of the equivariant quantum cohomology of the flag variety. Parabolic analogues of quantum double Schubert polynomials are introduced and shown to represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. This establishes a new method for computing equivariant Gromov-Witten invariants for partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.

References

D. Anderson, Double Schubert polynomials and double Schubert varieties, preprint, 2006.
D. Anderson and L. Chen, personal communication, 2010.
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 a flag space, Funkcional. Anal. i Priložen. 7 (1973), no. 1, 64–65 (Russian). MR 0318166
Sara C. Billey, Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J. 96 (1999), no. 1, 205–224. MR 1663931, DOI 10.1215/S0012-7094-99-09606-0
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 51508, DOI 10.2307/1969728
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
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
William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998. Appendix J by the authors in collaboration with I. Ciocan-Fontanine. MR 1639468, DOI 10.1007/BFb0096380
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
W. Fulton and C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), no. 4, 641–661. MR 2072765, DOI 10.1090/S1056-3911-04-00365-0
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
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Anatol N. Kirillov and Toshiaki Maeno, Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, Discrete Math. 217 (2000), no. 1-3, 191–223 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766267, DOI 10.1016/S0012-365X(99)00263-0
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
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
Bumsig Kim, On equivariant quantum cohomology, Internat. Math. Res. Notices 17 (1996), 841–851. MR 1420551, DOI 10.1155/S1073792896000517
Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$, Adv. in Math. 62 (1986), no. 3, 187–237. MR 866159, DOI 10.1016/0001-8708(86)90101-5
Thomas Lam and Mark Shimozono, Quantum cohomology of $G/P$ and homology of affine Grassmannian, Acta Math. 204 (2010), no. 1, 49–90. MR 2600433, DOI 10.1007/s11511-010-0045-8
Alain Lascoux and Marcel-Paul Schützenberger, Symmetry and flag manifolds, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 118–144. MR 718129, DOI 10.1007/BFb0063238
I. G. Macdonald, Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991) London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 73–99. MR 1161461
Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours Spécialisés [Specialized Courses], 3. MR 1852463
Leonardo Constantin Mihalcea, On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms, Duke Math. J. 140 (2007), no. 2, 321–350. MR 2359822, DOI 10.1215/S0012-7094-07-14024-9
Leonardo Constantin Mihalcea, Giambelli formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Amer. Math. Soc. 360 (2008), no. 5, 2285–2301. MR 2373314, DOI 10.1090/S0002-9947-07-04245-6
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
D. Peterson, Lecture notes at MIT, 1997.
Shawn Allen Robinson, Equivariant Schubert calculus, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–The University of North Carolina at Chapel Hill. MR 2701946