Duke Mathematical Journal

On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms

Leonardo Constantin Mihalcea

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X=G/P. Using this formula, we give an effective algorithm to compute the 3-point, genus zero, equivariant Gromov-Witten invariants on X, which are the structure constants of its equivariant quantum cohomology algebra

Article information

Source
Duke Math. J., Volume 140, Number 2 (2007), 321-350.

Dates
First available in Project Euclid: 18 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1192715422

Digital Object Identifier
doi:10.1215/S0012-7094-07-14024-9

Mathematical Reviews number (MathSciNet)
MR2359822

Zentralblatt MATH identifier
1135.14042

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14N15: Classical problems, Schubert calculus 57R91: Equivariant algebraic topology of manifolds 05E99: None of the above, but in this section

Citation

Mihalcea, Leonardo Constantin. On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms. Duke Math. J. 140 (2007), no. 2, 321--350. doi:10.1215/S0012-7094-07-14024-9. https://projecteuclid.org/euclid.dmj/1192715422


Export citation

References

  • A. Amarzaya and M. Guest, Gromov-Witten invariats of flag manifolds, via $D$-modules, J. London Math. Soc. (2) 72 (2005), 121--136.
  • A. Arabia, Cohomologie $T$-équivariante de la variété de drapeaux d'un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), 129--165.
  • A. Astashkevich and V. Sadov, Quantum cohomology of partial flag manifolds $F_n_1,\ldots ,n_k$, Comm. Math. Phys. 170 (1995), 503--528.
  • I. N. BernšTeĭN, I. M. Gel'Fand [Gelfand], and S. I. Gel'Fand [Gelfand], Schubert cells, and the cohomology of the spaces $G/P$ (in Russian), Uspehi Mat. Nauk 28, no. 3 (1973), 3--26.
  • A. Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), 289--305.
  • A. Bertram, I. Ciocan-Fontanine, and W. Fulton, Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), 728--746.
  • S. C. Billey, Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J. 96 (1999), 205--224.
  • A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
  • N. Bourbaki, Éléments de mathématique, fasc. 34: Groupes et algèbres de Lie, chapitres 4--6, Masson, Paris, 1981.
  • M. Brion, ``Equivariant cohomology and equivariant intersection theory'' in Representation Theories and Algebraic Geometry (Montréal, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad. Publ., Dordrecht, Netherlands, 1998, 1--37.
  • A. S. Buch, Quantum cohomology of Grassmannians, Compositio Math. 137 (2003), 227--235.
  • —, Quantum cohomology of the partial flag manifolds, Trans. Amer. Math. Soc. 357 (2005), 443--458.
  • A. S. Buch, A. Kresch, and H. Tamvakis, Gromov-Witten invariants on Grassmannians, J. Amer. Math. Soc. 16 (2003), 901--915.
  • L. Chen, Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), 1--34.
  • I. Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 1995, no. 6, 263--277.
  • —, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), 485--524.
  • I. Coskun, A Littlewood-Richardson rule for two-step flag varieties, preprint, 2006.
  • D. Edidin and W. Graham, Equivariant intersection theory, with appendix ``The Chow ring of $\mathcalM_2$'' by A. Vistoli, Invent. Math. 131 (1998), 595--634.
  • S. Fomin, personal communication, Nov. 2004.
  • S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565--596.
  • W. Fulton and R. Pandharipande, ``Notes on stable maps and quantum cohomology'' in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997, 45--96.
  • W. Fulton and C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), 641--661.
  • A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 1996, no. 13, 613--663.
  • —, ``Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture'' in Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser. 2 180, Amer. Math. Soc., Providence, 1997, 103--115.
  • —, ``A mirror theorem for toric complete intersections'' in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math. 160, Birkhäuser, Boston, 1998, 141--175.
  • A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), 609--641.
  • M. Goreski, R. Kottwitz, and R. Macpherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25--83.
  • W. Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), 599--614.
  • M. A. Guest, Quantum cohomology via $D$-modules, Topology 44 (2005), 263--281.
  • J. E. Humphreys, Linear Algebraic Groups, Grad. Texts in Math. 21, Springer, New York, 1975.
  • —, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge Univ. Press, Cambridge, 1990.
  • D. Husemoller, Fibre Bundles, 2nd ed., Grad. Texts in Math. 20, Springer, New York, 1975.
  • D. Joe and B. Kim, Equivariant mirrors and the Virasoro conjecture for flag manifolds, Int. Math. Res. Not. 2003, no. 15, 859--882.
  • B. Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices 1995, no. 1, 1--15.
  • —, On equivariant quantum cohomology, Internat. Math. Res. Notices 1996, no. 17, 841--851.
  • —, Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), 129--148.
  • B. Kim and R. Pandharipande, ``The connectedness of the moduli space of maps to homogeneous spaces'' in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 187--201.
  • S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287--297.
  • A. Knutson, A Schubert calculus recurrence from the noncomplex $W$-action on $G/B$, preprint,\arxivmath/0306304v1[math.CO]
  • A. Knutson and T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), 221--260.
  • M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525--562.
  • B. Kostant and S. Kumar, The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$, Adv. in Math. 62 (1986), 187--237.
  • A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom. 12 (2003), 777--810.
  • S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math. 204, Birkhäuser, Boston, 2002.
  • V. Lakshmibai and N. Gonciulea, Flag Varieties, Actualités Math., Hermann, Paris, 2001.
  • A.-L. Mare, ``On the theorem of Kim concerning $QH^*(G/B)$'' in Integrable Systems, Topology, and Physics (Tokyo, 2000), Contemp. Math. 309, Amer. Math. Soc., Providence, 2002, 151--163.
  • —, Polynomial representatives of Schubert classes in $QH^*(G/B)$, Math. Res. Lett. 9 (2002), 757--769.
  • —, On the Dubrovin connection for $G/B$, preprint,\arxivmath/0311320v2[math.DG]
  • L. C. Mihalcea, Equivariant quantum Schubert calculus, Adv. Math. 203 (2006), 1--33.
  • —, Positivity in equivariant quantum Schubert calculus, Amer. J. Math. 128 (2006), 787--803.
  • —, Equivariant quantum cohomology of homogeneous spaces, Ph.D. dissertation, University of Michigan, Ann Arbor, Mich., 2005.
  • D. Peterson, lectures, Massachusetts Insitute of Technology, Cambridge, Mass., 1997.
  • A. Postnikov, Affine approach to quantum Schubert calculus, Duke Math. J. 128 (2005), 473--509.
  • K. Rietsch, A mirror symmetric construction of $QH^*_T(G/P)_(q)$, preprint,\arxivmath/0511124v1[math.AG]
  • S. Robinson, A Pieri-type formula for $H\sp*\sb T(\rm SL\sb n(\BbbC)/B)$, J. Algebra 249 (2002), 38--58.
  • J. F. Thomsen, Irreducibility of $\overlineM\sb 0,n(G/P,\beta)$, Internat. J. Math. 9 (1998), 367--376.
  • A. Vistoli, The Chow ring of $\mathcalM_2$, appendix to Equivariant intersection theory by D. Edidin and W. Graham, Invent. Math. 131 (1998), 635--644.
  • E. Witten, ``The Verlinde algebra and the cohomology of the Grassmannian'' in Geometry, Topology, and Physics (Cambridge, Mass., 1993), Int. Press, Cambridge, Mass., 1995, 357--422.
  • C. T. Woodward, On D. Peterson's comparison formula for Gromov-Witten invariants of $G/P$, Proc. Amer. Math. Soc. 133 (2005), 1601--1609.