Double Schubert polynomials and degeneracy loci for the classical groups

Kresch, Andrew; Tamvakis, Harry

doi:10.5802/aif.1931

Kresch, Andrew; Tamvakis, Harry

Annales de l'Institut Fourier, Volume 52 (2002) no. 6, p. 1681-1727

Abstract
Résumé

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.

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.

MR 1952528 | Zbl 1059.14063

DOI : https://doi.org/10.5802/aif.1931
Classification: 14M15, 14C17, 05E15
Keywords: degeneracy loci, Schubert polynomials

@article{AIF_2002__52_6_1681_0,
     author = {Kresch, Andrew and Tamvakis, Harry},
     title = {Double Schubert polynomials and degeneracy loci for the classical groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
     pages = {1681-1727},
     doi = {10.5802/aif.1931},
     zbl = {1059.14063},
     mrnumber = {1952528},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_6_1681_0}
}

Kresch, Andrew; Tamvakis, Harry. Double Schubert polynomials and degeneracy loci for the classical groups. Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1681-1727. doi : 10.5802/aif.1931. http://www.numdam.org/item/AIF_2002__52_6_1681_0/

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