Double Schubert polynomials and degeneracy loci for the classical groups

Andrew Kresch[1]; Harry Tamvakis[2]

  • [1] University of Pennsylvania, Department of Mathematics, Philadelphia PA 19104-6395 (USA)
  • [2] Brandeis University, Department of Mathematics, Waltham MA 02454-9110 (USA)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 6, page 1681-1727
  • ISSN: 0373-0956

Abstract

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We propose a theory of double Schubert polynomials P w ( X , Y ) for the Lie types B , C , D which naturally extends the family of Lascoux and Schützenberger in type A . These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When w is a maximal Grassmannian element of the Weyl group, P w ( X , Y ) can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type A formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows there are no Chern class formulas for degeneracy loci of “isotropic morphisms” of bundles.

How to cite

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Kresch, Andrew, and Tamvakis, Harry. "Double Schubert polynomials and degeneracy loci for the classical groups." Annales de l’institut Fourier 52.6 (2002): 1681-1727. <http://eudml.org/doc/116024>.

@article{Kresch2002,
abstract = {We propose a theory of double Schubert polynomials $P_w(X,Y)$ for the Lie types $B$, $C$, $D$ which naturally extends the family of Lascoux and Schützenberger in type $A$. These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When $w$ is a maximal Grassmannian element of the Weyl group, $P_w(X,Y)$ can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type $A$ formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows there are no Chern class formulas for degeneracy loci of “isotropic morphisms” of bundles.},
affiliation = {University of Pennsylvania, Department of Mathematics, Philadelphia PA 19104-6395 (USA); Brandeis University, Department of Mathematics, Waltham MA 02454-9110 (USA)},
author = {Kresch, Andrew, Tamvakis, Harry},
journal = {Annales de l’institut Fourier},
keywords = {degeneracy loci; Schubert polynomials; determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes},
language = {eng},
number = {6},
pages = {1681-1727},
publisher = {Association des Annales de l'Institut Fourier},
title = {Double Schubert polynomials and degeneracy loci for the classical groups},
url = {http://eudml.org/doc/116024},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Kresch, Andrew
AU - Tamvakis, Harry
TI - Double Schubert polynomials and degeneracy loci for the classical groups
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 6
SP - 1681
EP - 1727
AB - We propose a theory of double Schubert polynomials $P_w(X,Y)$ for the Lie types $B$, $C$, $D$ which naturally extends the family of Lascoux and Schützenberger in type $A$. These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When $w$ is a maximal Grassmannian element of the Weyl group, $P_w(X,Y)$ can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type $A$ formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows there are no Chern class formulas for degeneracy loci of “isotropic morphisms” of bundles.
LA - eng
KW - degeneracy loci; Schubert polynomials; determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes
UR - http://eudml.org/doc/116024
ER -

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