Twisted action of the symmetric group on the cohomology of a flag manifold

• Autorzy: Alain Lascoux , Bernard Leclerc , Jean-Yves Thibon
• Języki publikacji: EN

Abstrakty

EN

Classes dual to Schubert cycles constitute a basis on the cohomology ring of the flag manifold F, self-adjoint up to indexation with respect to the intersection form. Here, we study the bilinear form (X,Y) :=〈X·Y, c(F)〉 where X,Y are cocycles, c(F) is the total Chern class of F and〈,〉 is the intersection form. This form is related to a twisted action of the symmetric group of the cohomology ring, and to the degenerate affine Hecke algebra. We give a distinguished basis for this form, which is a deformation of the usual basis of Schubert polynomials, and apply it to the computation of the Schubert cycle expansions of Chern classes of flag manifolds.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1996
• Tom: 36
• Numer: 1
• Strony: 111-124

Daty

wydano

1996

Twórcy

autor

Alain Lascoux

• L.I.T.P., Université Paris 7, 2, Place Jussieu, 75251 Paris Cedex 05, France

autor

Bernard Leclerc

• L.I.T.P., Université Paris 7, 2, Place Jussieu, 75251 Paris Cedex 05, France

autor

Jean-Yves Thibon

• Institut Gaspard Monge, Université de Marne-la-Vallée, 2, rue de la Butte-Verte, 93166 Noisy-le-Grand Cedex, France

Bibliografia

• [1] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and the cohomology of the spaces G/P, Russian Math. Surveys 28 (1973), 1-26.
• [2] I. V. Cherednik, On R-matrix quantization of formal loop groups, in: Group theoretical methods in physics, Vol. II (Yurmala, 1985), 161-180, VNU Sci. Press, Utrecht, 1986.
• [3] I. V. Cherednik, Quantum groups as hidden symmetries of classic representation theory, in: Differential geometric methods in theoretical physics (A. I. Solomon ed.), World Scientific, Singapore, 1989, 47-54.
• [4] M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88.
• [5] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301.
• [6] G. Duchamp, D. Krob, A. Lascoux, B. Leclerc, T. Scharf and J.-Y. Thibon, Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras, Publ. Res. Inst. Math. Sci. 31 (1995), 179-201.
• [7] W. Fulton, Schubert varieties in flag bundles for the Classical Groups, preprint, University of Chicago, 1994; to appear in: Proceedings of the Conference in Honor of Hirzebruch's 65th Birthday, Bar Ilan, 1993.
• [8] F. Hirzebruch, Topological methods in algebraic geometry, Springer, Berlin, 1966.
• [9] A. Kerber, A. Kohnert and A. Lascoux, SYMMETRICA, an object oriented computer algebra system for the symmetric group, J. Symbolic Comput. 14 (1992), 195-203.
• [10] A. Lascoux, Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 393-398.
• [11] A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447-450.
• [12] A. Lascoux and M.-P. Schützenberger, Symmetrization operators on polynomial rings, Functional Anal. Appl. 21 (1987), 77-78.
• [13] G. Lusztig, Equivariant K-theory and representations of Hecke Algebras, Proc. Amer. Math. Soc. 94 (1985), 337-342.
• [14] I. G. Macdonald, Notes on Schubert polynomials, Publ. LACIM 6, UQAM, Montréal, 1991.
• [15] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci: the $\tilde Q$-polynomials approach, Max-Planck-Institut für Mathematik Preprint 1994; to appear in Compositio Math.
• [16] S. Veigneau, SP, a Maple package for Schubert polynomials, Université de Marne-la-Vallée, 1994.
• [17] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.