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

Alain Lascoux; Bernard Leclerc; Jean-Yves Thibon

Banach Center Publications (1996)

  • Volume: 36, Issue: 1, page 111-124
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Lascoux, Alain, Leclerc, Bernard, and Thibon, Jean-Yves. "Twisted action of the symmetric group on the cohomology of a flag manifold." Banach Center Publications 36.1 (1996): 111-124. <http://eudml.org/doc/208575>.

@article{Lascoux1996,
abstract = {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.},
author = {Lascoux, Alain, Leclerc, Bernard, Thibon, Jean-Yves},
journal = {Banach Center Publications},
keywords = {cohomology ring; action of the symmetric group; degenerate affine Hecke algebra; variety of complete flags; Schubert polynomials; Schubert cycles},
language = {eng},
number = {1},
pages = {111-124},
title = {Twisted action of the symmetric group on the cohomology of a flag manifold},
url = {http://eudml.org/doc/208575},
volume = {36},
year = {1996},
}

TY - JOUR
AU - Lascoux, Alain
AU - Leclerc, Bernard
AU - Thibon, Jean-Yves
TI - Twisted action of the symmetric group on the cohomology of a flag manifold
JO - Banach Center Publications
PY - 1996
VL - 36
IS - 1
SP - 111
EP - 124
AB - 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.
LA - eng
KW - cohomology ring; action of the symmetric group; degenerate affine Hecke algebra; variety of complete flags; Schubert polynomials; Schubert cycles
UR - http://eudml.org/doc/208575
ER -

References

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  1. [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. [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. [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. [4] M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. Zbl0312.14009
  5. [5] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301. Zbl0269.22010
  6. [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. Zbl0835.05085
  7. [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. [8] F. Hirzebruch, Topological methods in algebraic geometry, Springer, Berlin, 1966. Zbl0138.42001
  9. [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. Zbl0823.20015
  10. [10] A. Lascoux, Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 393-398. Zbl0495.14032
  11. [11] A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447-450. 
  12. [12] A. Lascoux and M.-P. Schützenberger, Symmetrization operators on polynomial rings, Functional Anal. Appl. 21 (1987), 77-78. 
  13. [13] G. Lusztig, Equivariant K-theory and representations of Hecke Algebras, Proc. Amer. Math. Soc. 94 (1985), 337-342. Zbl0571.22014
  14. [14] I. G. Macdonald, Notes on Schubert polynomials, Publ. LACIM 6, UQAM, Montréal, 1991. 
  15. [15] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci: the Q ˜ -polynomials approach, Max-Planck-Institut für Mathematik Preprint 1994; to appear in Compositio Math. 
  16. [16] S. Veigneau, SP, a Maple package for Schubert polynomials, Université de Marne-la-Vallée, 1994. 
  17. [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. Zbl0152.46301

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