Skip to main content
SpringerLink
Log in

Equivariant Schubert calculus

  • Published:
Arkiv för Matematik

Abstract

We describe T-equivariant Schubert calculus on G(k,n), T being an n-dimensional torus, through derivations on the exterior algebra of a free A-module of rank n, where A is the T-equivariant cohomology of a point. In particular, T-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Quart. 2 (2006), 699–717).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brion, M., Equivariant cohomology and equivariant intersection theory, in Representation Theories and Algebraic Geometry (Montreal, QC, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, pp. 1–37, Kluwer, Dordrecht, 1997.

    Google Scholar 

  2. Cordovez, J., Gatto, L. and Santiago, L., Newton binomial formulas in Schubert calculus, Rev. Mat. Complut.22 (2008), 129–152.

    MathSciNet  Google Scholar 

  3. Fulton, W., Intersection Theory, Springer, Berlin, 1998.

    MATH  Google Scholar 

  4. Fulton, W., Equivariant Intersection Theory, Eilenberg Lectures at Columbia University, New York, 2007, notes by D. Anderson available at http://www.math.lsa.umich.edu/dandersen/eilenberg/.

  5. Gatto, L., Schubert calculus via Hasse–Schmidt derivations, Asian J. Math.9 (2005), 315–322.

    MATH  MathSciNet  Google Scholar 

  6. Gatto, L., Schubert Calculus: An Algebraic Introduction, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 2005.

    MATH  Google Scholar 

  7. Gatto, L. and Santiago, T., Schubert calculus on a Grassmann algebra, to appear in Canad. Math. Bull.

  8. Goresky, M., Kottwitz, R. and MacPherson, R., Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math.131 (1998), 25–83.

    Article  MATH  MathSciNet  Google Scholar 

  9. Knutson, A. and Tao, T., Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J.119 (2003), 221–260.

    Article  MATH  MathSciNet  Google Scholar 

  10. Lakshmibai, V., Raghavan, K. N. and Sankaran, P., Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Quart.2 (2006), 699–717.

    MATH  MathSciNet  Google Scholar 

  11. Laksov, D., The formalism of equivariant Schubert calculus, Preprint, 2006.

  12. Laksov, D., Schubert calculus and equivariant cohomology of Grassmannians, Adv. Math.217 (2008), 1869–1888.

    Article  MATH  MathSciNet  Google Scholar 

  13. Laksov, D. and Thorup, A., A determinantal formula for the exterior powers of the polynomial ring, Indiana Univ. Math. J.56 (2007), 825–845.

    Article  MATH  MathSciNet  Google Scholar 

  14. Laksov, D. and Thorup, A., Schubert calculus on Grassmannians and exterior products, Indiana Univ. Math. J.58 (2009), 283–300.

    Article  MATH  MathSciNet  Google Scholar 

  15. Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd ed., Clarendon, Oxford, 1979.

    MATH  Google Scholar 

  16. Mihalcea, L. C., Equivariant quantum Schubert calculus, Adv. Math.203 (2006), 1–33.

    Article  MATH  MathSciNet  Google Scholar 

  17. Mihalcea, L. C., Giambelli formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Amer. Math. Soc.360 (2008), 2285–2301.

    Article  MATH  MathSciNet  Google Scholar 

  18. Santiago, T., Schubert Calculus on a Grassmann Algebra. Ph.D. Thesis, Politecnico di Torino, Torino, 2006.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, IT-10129, Torino, Italy

    Letterio Gatto

  2. Instituto de Matemática, Universidade Federal da Bahia, BR-40170-110, Salvador, Bahia, Brazil

    Taíse Santiago

Authors
  1. Letterio Gatto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taíse Santiago
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Letterio Gatto.

Additional information

Work partially sponsored by PRIN “Geometria sulle Varietà Algebriche” (Coordinatore Alessandro Verra), INDAM-GNSAGA, Scuola di Dottorato (ScuDo) del Politecnico di Torino, FAPESB proc. n. 8057/2006 and CNPq proc. n. 350259/2006-2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gatto, L., Santiago, T. Equivariant Schubert calculus. Ark Mat 48, 41–55 (2010). https://doi.org/10.1007/s11512-009-0093-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11512-009-0093-5

Keywords

Search

Navigation