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Pieri’S Formula Via Explicit Rational Equivalence

Published online by Cambridge University Press:  20 November 2018

Frank Sottile
Frank Sottile*
Affiliation:
Department of Mathematics University of Toronto 100 St. George Street Toronto, Ontario
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Abstract

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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.

Keywords

14M1505E10Pieri’s formularational equivalenceGrassmannianSchensted insertion
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 6 , 01 December 1997 , pp. 1281 - 1298
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Allgower, E. and Georg, K., Numerical Continuation Methods, An Introduction. Springer Ser. Comput. Math. 13, Springer-Verlag, 1990.Google Scholar
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), 1–26.Google Scholar
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. 1–23.Google Scholar
4. Demazure, M., Désingularization des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7(1974), 53–88.Google Scholar
5. Fulton, W., Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge, 1996.Google Scholar
6. Fulton, W. and Harris, J., Representation Theory. Graduate Texts in Math 129, Springer-Verlag, 1991.Google Scholar
7. Griffiths, P. and Harris, J., Principles of Algebraic Geometry. J. Wiley and Sons, New York, 1978.Google Scholar
8. Hiller, H., The Geometry of Coxeter Groups. Pitman Res. Notes Math. Ser. 54,Pitman, Boston, MA, 1982.Google Scholar
9. Hodge, W.V.D., The intersection formula for a Grassmannian variety. J. London Math. Soc. 17(1942), 48–64.Google Scholar
10. Huber, B., Sottile, F., and Sturmfels, B., Numerical Schubert calculus. 1997.Google Scholar
11. Kleiman, S., The transversality of a general translate. Compositio Math. 28(1974), 287–297.Google Scholar
12. Laksov, D., Algebraic cycles in Grassmann varieties. Adv. Math. 9(1972), 267–295.Google Scholar
13. Macdonald, I.G., Symmetric Functions and Hall Polynomials. 2nd edn, Oxford University Press, New York, 1995.Google Scholar
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. 125–177.Google Scholar
15. Pragacz, P. and Ratajski, J., Pieri type formula for isotropic Grassmannians; the operator approach. Manuscripta Math. 79(1993), 127–151.Google Scholar
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), 1035–1040.Google Scholar
17. Pragacz, P., Pieri-type formula for Lagrangian and odd orthogonal Grassmannians. J. Reine Angew. Math. 476(1996), 143–189.Google Scholar
18. Pragacz, P., A Pieri-type theorem for even orthogonal Grassmannians. Max-Planck Institut preprint, 1996.Google Scholar
19. Sagan, B., The Symmetric Group; Representations, Combinatorics, Algorithms & Symmetric Functions. Wadsworth & Brooks/Cole, 1991.Google Scholar
20. Samuel, P., Méthodes d’Algèbre Abstraite en Géométrie Algébrique. Seconde édition, Ergeb. Math. Grenzgeb., Springer-Verlag, 1967.Google Scholar
21. Schensted, C., Longest increasing and decreasing subsequence, Can. J. Math. 13(1961), 179–191.Google Scholar
22. Sottile, F., Pieri's formula for flag manifolds and Schubert polynomials. Ann. Inst. Fourier (Grenoble) 46(1996), 89–110.Google Scholar
23. Sottile, F., Enumerative geometry for the real Grassmannian of lines in projective space. DukeMath. J. 87(1997), 59–85.Google Scholar
24. Sottile, F., Real enumerative geometry and effective algebraic equivalence. J. Pure Appl. Algebra 117/118(1997), 601–615.Google Scholar