Skip to main content Accessibility help
×
Home

Pieri’S Formula Via Explicit Rational Equivalence

  • Frank Sottile (a1)

Abstract

Pieri’s formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri’s formula using Schensted insertion.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Pieri’S Formula Via Explicit Rational Equivalence
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Pieri’S Formula Via Explicit Rational Equivalence
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Pieri’S Formula Via Explicit Rational Equivalence
      Available formats
      ×

Copyright

References

Hide All
1. Allgower, E. and Georg, K., Numerical Continuation Methods, An Introduction. Springer Ser. Comput. Math. 13, Springer-Verlag, 1990.
2. Bernstein, I.N., Gelfand, I.M., and Gelfand, S.I., Schubert cells and cohomology of the spaces G ÛP. Russian Math. Surveys 28(1973), 126.
3. Chevalley, C., Sur les décompositions cellulaires des espaces G/B. Proc. Sympos. Pure Math. (1) 56, Algebraic Groups and their Generalizations: Classical Methods, Amer. Math. Soc., Providence, RI, 1994. 123.
4. Demazure, M., Désingularization des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7(1974), 5388.
5. Fulton, W., Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge, 1996.
6. Fulton, W. and Harris, J., Representation Theory. Graduate Texts in Math 129, Springer-Verlag, 1991.
7. Griffiths, P. and Harris, J., Principles of Algebraic Geometry. J. Wiley and Sons, New York, 1978.
8. Hiller, H., The Geometry of Coxeter Groups. Pitman Res. Notes Math. Ser. 54,Pitman, Boston, MA, 1982.
9. Hodge, W.V.D., The intersection formula for a Grassmannian variety. J. London Math. Soc. 17(1942), 4864.
10. Huber, B., Sottile, F., and Sturmfels, B., Numerical Schubert calculus. 1997.
11. Kleiman, S., The transversality of a general translate. Compositio Math. 28(1974), 287297.
12. Laksov, D., Algebraic cycles in Grassmann varieties. Adv. Math. 9(1972), 267295.
13. Macdonald, I.G., Symmetric Functions and Hall Polynomials. 2nd edn, Oxford University Press, New York, 1995.
14. Pragacz, P., Symmetric polynomials and divided differences in formulas of intersection theory. In: Parameter Spaces 36, Banach Center Publications, Banach Center workshop, 1994. Institute of Mathematics, Polish Academy of Sciences, 1996. 125177.
15. Pragacz, P. and Ratajski, J., Pieri type formula for isotropic Grassmannians; the operator approach. Manuscripta Math. 79(1993), 127151.
16. Pragacz, P., Pieri-type formula for SP(2m)/P and SO(2m + 1)/P. C. R. Acad. Sci. Paris Sér. I Math. 317(1993), 10351040.
17. Pragacz, P., Pieri-type formula for Lagrangian and odd orthogonal Grassmannians. J. Reine Angew. Math. 476(1996), 143189.
18. Pragacz, P., A Pieri-type theorem for even orthogonal Grassmannians. Max-Planck Institut preprint, 1996.
19. Sagan, B., The Symmetric Group; Representations, Combinatorics, Algorithms & Symmetric Functions. Wadsworth & Brooks/Cole, 1991.
20. Samuel, P., Méthodes d’Algèbre Abstraite en Géométrie Algébrique. Seconde édition, Ergeb. Math. Grenzgeb., Springer-Verlag, 1967.
21. Schensted, C., Longest increasing and decreasing subsequence, Can. J. Math. 13(1961), 179191.
22. Sottile, F., Pieri's formula for flag manifolds and Schubert polynomials. Ann. Inst. Fourier (Grenoble) 46(1996), 89110.
23. Sottile, F., Enumerative geometry for the real Grassmannian of lines in projective space. DukeMath. J. 87(1997), 5985.
24. Sottile, F., Real enumerative geometry and effective algebraic equivalence. J. Pure Appl. Algebra 117/118(1997), 601615.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Pieri’S Formula Via Explicit Rational Equivalence

  • Frank Sottile (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed