Double Schubert polynomials and degeneracy loci for the classical groups

Kresch, Andrew; Tamvakis, Harry

doi:10.5802/aif.1931

Double Schubert polynomials and degeneracy loci for the classical groups
[Polynômes de Schubert doubles et lieux de dégénérescence pour les groupes classiques]

Kresch, Andrew ¹ ; Tamvakis, Harry 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, Tome 52 (2002) no. 6, pp. 1681-1727.

Résumé
Abstract

Nous proposons une théorie des polynômes de Schubert doubles $P_{w} (X, Y)$ pour les types de Lie $B, C, D$ qui étend naturellement la famille de Lascoux et Schützenberger pour le type $A$ . Ces polynômes possèdent des propriétés de positivité, d’orthogonalité et de stabilité, et représentent les classes des variétés de Schubert et des lieux de dégénérescence des fibrés vectoriels. Quand $w$ est un élément grassmannien maximal du groupe de Weyl, $P_{w} (X, Y)$ s’exprime en termes de déterminants du type de Schur et de pfaffiens, de manière analogue à la formule de Kempf et Laksov pour le type $A$ . Un exemple, motivé par la cohomologie quantique, montre qu’aucune formule dans les classes de Chern ne décrit les lieux de dégénérescence des “morphismes isotropes” des fibrés.

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.

MR Zbl

DOI : 10.5802/aif.1931

Classification : 14M15, 14C17, 05E15
Keywords: degeneracy loci, Schubert polynomials
Mot clés : lieux de dégénérescence, polynômes de Schubert

Affiliations des auteurs :

Kresch, Andrew ¹ ; Tamvakis, Harry ²

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

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     title = {Double {Schubert} polynomials and degeneracy loci for the classical groups},
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     pages = {1681--1727},
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Kresch, Andrew; Tamvakis, Harry. Double Schubert polynomials and degeneracy loci for the classical groups. Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1681-1727. doi : 10.5802/aif.1931. https://aif.centre-mersenne.org/articles/10.5802/aif.1931/

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