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Double Schubert polynomials and degeneracy loci for the classical groups


Kresch, A; Tamvakis, H (2002). Double Schubert polynomials and degeneracy loci for the classical groups. Annales de l'institut Fourier, 52(6):1681-1727.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:degeneracy loci, Schubert polynomials
Language:English
Date:2002
Deposited On:18 Feb 2010 11:26
Last Modified:05 Apr 2016 13:25
Publisher:Association des Annales de l'Institut Fourier
ISSN:0373-0956
Additional Information:© Association des Annales de l'institut Fourier, 2002, touts droits réservés
Official URL:http://aif.cedram.org/item?id=AIF_2002__52_6_1681_0
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1952528
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1059.14063
Permanent URL: http://doi.org/10.5167/uzh-21955

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