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Quantum double Schubert polynomials represent Schubert classes


Authors: Thomas Lam and Mark Shimozono
Journal: Proc. Amer. Math. Soc. 142 (2014), 835-850
MSC (2010): Primary 14N35; Secondary 14M15
DOI: https://doi.org/10.1090/S0002-9939-2013-11831-9
Published electronically: December 11, 2013
MathSciNet review: 3148518
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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.


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  • [And] D. Anderson, Double Schubert polynomials and double Schubert varieties, preprint, 2006.
  • [AC] D. Anderson and L. Chen, personal communication, 2010.
  • [AS] 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 (96g:58027)
  • [BGG] I. N. Bernšteĭn, I. M. Gelfand, and S. I. Gelfand, Schubert cells and the cohomology of a flag space, Funkcional. Anal. i Priložen. 7 (1973), no. 1, 64-65 (Russian). MR 0318166 (47 #6713)
  • [Bi] Sara C. Billey, Kostant polynomials and the cohomology ring for $ G/B$, Duke Math. J. 96 (1999), no. 1, 205-224. MR 1663931 (2000a:14060), https://doi.org/10.1215/S0012-7094-99-09606-0
  • [Bo] 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 (14,490e)
  • [Che] C. Chevalley, Sur les décompositions cellulaires des espaces $ G/B$. With a foreword by Armand Borel (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1-23 (French). MR 1278698 (95e:14041). .
  • [Cio] Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263-277. MR 1344348 (96h:14071), https://doi.org/10.1155/S1073792895000213
  • [Cio2] Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485-524. MR 1695799 (2000d:14058), https://doi.org/10.1215/S0012-7094-99-09815-0
  • [CF] I. Ciocan-Fontanine and W. Fulton, Quantum double Schubert polynomials. Appendix J in Schubert Varieties and Degeneracy Loci, Lecture Notes in Math. 1689 (1998), 134-138. MR 1639468 (99m:14092)
  • [FGP] Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565-596. MR 1431829 (98d:14063), https://doi.org/10.1090/S0894-0347-97-00237-3
  • [FW] W. Fulton and C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), no. 4, 641-661. MR 2072765 (2005d:14078), https://doi.org/10.1090/S1056-3911-04-00365-0
  • [GK] Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609-641. MR 1328256 (96c:58027)
  • [Hum] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [KM] Anatol N. Kirillov and Toshiaki Maeno, Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, Formal power series and algebraic combinatorics (Vienna, 1997), Discrete Math. 217 (2000), no. 1-3, 191-223 (English, with English and French summaries). MR 1766267 (2001f:05161), https://doi.org/10.1016/S0012-365X(99)00263-0
  • [Kim] 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 (2001c:14081), https://doi.org/10.2307/121021
  • [Kim2] 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 (96c:58028), https://doi.org/10.1155/S1073792895000018
  • [Kim3] Bumsig Kim, On equivariant quantum cohomology, Internat. Math. Res. Notices 17 (1996), 841-851. MR 1420551 (98h:14013), https://doi.org/10.1155/S1073792896000517
  • [KK] 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 (88b:17025b), https://doi.org/10.1016/0001-8708(86)90101-5
  • [LamSh] 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 (2011h:14082), https://doi.org/10.1007/s11511-010-0045-8
  • [LS] 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 (85e:14073), https://doi.org/10.1007/BFb0063238
  • [Mac] 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 (93d:05159)
  • [Man] Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, translated from the 1998 French original by John R. Swallow. SMF/AMS Texts and Monographs, vol. 6, Cours Spécialisés [Specialized Courses], 3, American Mathematical Society, Providence, RI, and Société Mathématique de France, Paris, 2001. MR 1852463 (2002h:05161)
  • [Mi] 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 (2008j:14106), https://doi.org/10.1215/S0012-7094-07-14024-9
  • [Mi2] 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 (2009e:14099), https://doi.org/10.1090/S0002-9947-07-04245-6
  • [Mo] D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253-286. MR 0106911 (21 #5641)
  • [Pet] D. Peterson, Lecture notes at MIT, 1997.
  • [Rob] 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

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Additional Information

Thomas Lam
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
Email: tfylam@umich.edu

Mark Shimozono
Affiliation: Department of Mathematics, MC0151, 460 McBryde Hall, Virginia Tech, 225 Stanger Street, Blacksburg, Virginia 24061
Email: mshimo@vt.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11831-9
Received by editor(s): October 22, 2011
Received by editor(s) in revised form: April 17, 2012
Published electronically: December 11, 2013
Additional Notes: The first author was supported by NSF grant DMS-0901111 and by a Sloan Fellowship.
The second author was supported by NSF DMS-0652641 and DMS-0652648.
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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