Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Giambelli formulae for the equivariant quantum cohomology of the Grassmannian


Author: Leonardo Constantin Mihalcea
Journal: Trans. Amer. Math. Soc. 360 (2008), 2285-2301
MSC (2000): Primary 14N35; Secondary 05E05, 14F43
DOI: https://doi.org/10.1090/S0002-9947-07-04245-6
Published electronically: December 11, 2007
MathSciNet review: 2373314
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.


References [Enhancements On Off] (What's this?)

  • 1. A. Arabia.
    Cohomologie $ {T}$-èquivariante de la variété de drapeaux d'un groupe de Kac-Moody.
    Bull. Soc. Math. France, 117(2):129-165, 1989. MR 1015806 (90i:32042)
  • 2. A. Astashkevich and V. Sadov.
    Quantum cohomology of partial flag manifolds $ {F}_{n_1,...,n_k}$.
    Commun. Math. Phys., 170:503-528, 1995. MR 1337131 (96g:58027)
  • 3. A. Bertram.
    Quantum Schubert Calculus.
    Adv. Math., 128(2):289-305, 1997. MR 1454400 (98j:14067)
  • 4. A. Bertram, I. Ciocan-Fontanine, and W. Fulton.
    Quantum multiplication of Schur polynomials.
    Journal of Algebra, 219(2):728-746, 1999. MR 1706853 (2000k:14042)
  • 5. L. Biedenharn and J. Louck.
    A new class of symmetric polynomials defined in terms of tableaux.
    Advances in Applied Math., 10:396-438, 1989. MR 1023942 (91c:05189)
  • 6. S. C. Billey.
    Kostant polynomials and the cohomology ring of $ {G/B}$.
    Duke Math. J., 96:205-224, 1999. MR 1663931 (2000a:14060)
  • 7. M. Brion.
    Poincaré duality and equivariant (co)homology.
    Michigan Math. J. - special volume in honor of William Fulton, 48:77-92, 2000. MR 1786481 (2001m:14032)
  • 8. A. S. Buch.
    Quantum cohomology of the Grassmannians.
    Compositio Math., 137(2):227-235, 2003. MR 1985005 (2004c:14105)
  • 9. W. Chen and J. D. Louck.
    The factorial Schur function.
    J. of Math. Phys., 34(9):4144-4160, 1993. MR 1233264 (95b:05210)
  • 10. D. Eisenbud.
    Commutative Algebra. With a view towards Algebraic Geometry.
    Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • 11. W. Fulton.
    Intersection Theory.
    Springer-Verlag, 2nd edition, 1998. MR 1644323 (99d:14003)
  • 12. A. Givental.
    Equivariant Gromov-Witten invariants.
    IMRN, (13):613-663, 1996. MR 1408320 (97e:14015)
  • 13. A. Givental and B. Kim.
    Quantum cohomology of flag manifolds and Toda lattices.
    Comm. Math. Phys., 168:609-641, 1995. MR 1328256 (96c:58027)
  • 14. I. Goulden and C. Greene.
    A new tableaux representation for supersymmetric Schur functions.
    Journal of Algebra, 170:687-703, 1994. MR 1302864 (96f:05187)
  • 15. B. Kim.
    Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings.
    IMRN, (1):1-15, 1995. MR 1317639 (96c:58028)
  • 16. B. Kim.
    On equivariant quantum cohomology.
    IMRN, (17):841-851, 1996. MR 1420551 (98h:14013)
  • 17. B. Kim.
    Quantum cohomology of flag manifolds $ {G/B}$ and quantum Toda lattices.
    Annals of Math., 149:129-148, 1999. MR 1680543 (2001c:14081)
  • 18. A. Knutson and T. Tao.
    Puzzles and equivariant cohomology of Grassmannians.
    Duke Math. J., 119(2):221-260, 2003. MR 1997946 (2006a:14088)
  • 19. V. Lakshmibai, K.N. Raghavan, and P. Sankaran.
    Equivariant Giambelli and determinantal restriction formulas for the Grassmannian.
    Pure Appl. Math. Q. 2(3):699-717, 2006. MR 2252114 (2007h:14084)
  • 20. A. Lascoux.
    Interpolation - lectures at Tianjin University.
    June 1996.
  • 21. I. G. Macdonald.
    Schur functions, theme and variations.
    Actes 28éme Séminaire Lotharingien, pages 5-29, 1992. MR 1308728 (95m:05245)
  • 22. I. G. Macdonald.
    Symmetric functions and Hall polynomials. With contributions by A. Zelevinsky.
    Oxford University Press, New York, 2nd edition, 1995. MR 1354144 (96h:05207)
  • 23. L. C. Mihalcea.
    Equivariant quantum Schubert Calculus.
    Adv. of Math., 203(1):1-33, 2006. MR 2231042 (2007c:14061)
  • 24. L. C. Mihalcea.
    Positivity in equivariant quantum Schubert calculus.
    Amer. J. of Math., 128(3):787-803, 2006. MR 2230925
  • 25. A. I. Molev.
    Factorial supersymmetric Schur functions and super Capelli identities.
    In Proc. of the AMS - Kirillov's seminar on representation theory, pages 109-137, Providence, RI, 1998. Amer. Math. Soc.
  • 26. A. I. Molev and B. Sagan.
    A Littlewood-Richardson rule for factorial Schur functions.
    Trans. of Amer. Math. Soc., 351(11):4429-4443, 1999. MR 1621694 (2000a:05212)
  • 27. A. Okounkov.
    Quantum immanants and higher Capelli identities.
    Transformations Groups, 1:99-126, 1996. MR 1390752 (97j:17010)
  • 28. A. Okounkov and O. Olshanski.
    Shifted Schur functions.
    St. Petersburg Math. J., 9(2), 1997.
    also available on ar$ \chi$iv: math: q-alg/9605042.
  • 29. A. Postnikov.
    Affine approach to quantum Schubert calculus.
    Duke Math. J., 128(3):473-509, 2003. MR 2145741 (2006e:05182)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14N35, 05E05, 14F43

Retrieve articles in all journals with MSC (2000): 14N35, 05E05, 14F43


Additional Information

Leonardo Constantin Mihalcea
Affiliation: Department of Mathematics, Florida State University, 208 Love Building, Tallahassee, Florida 32312
Address at time of publication: Department of Mathematics, Duke University, Box 90320, Durham, North Carolina 27708
Email: mihalcea@math.fsu.edu, lmihalce@math.duke.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04245-6
Received by editor(s): June 17, 2005
Received by editor(s) in revised form: November 9, 2005
Published electronically: December 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society