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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  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. 
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