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},
     journal = {Annales de l'Institut Fourier},
     pages = {1681--1727},
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     volume = {52},
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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. http://www.numdam.org/articles/10.5802/aif.1931/

Bibliographie
Cité par

[AC] E. Akyildiz; J. Carrell An algebraic formula for the Gysin homomorphism from $G / B$ to $G / P$ , Illinois J. Math, Volume 31 (1987) no. 2, pp. 312-320 | MR | Zbl

[Be] A. Bertram Quantum Schubert calculus, Adv. Math, Volume 128 (1997) no. 2, pp. 289-305 | DOI | MR | Zbl

[BGG] I. N. Bernstein; I. M. Gelfand; S. I. Gelfand Schubert cells and cohomology of the spaces $G / P$ , Russian Math. Surveys, Volume 28 (1973) no. 3, pp. 1-26 | DOI | MR | Zbl

[BH] S. Billey; M. Haiman Schubert polynomials for the classical groups, J. Amer. Math. Soc, Volume 8 (1995) no. 2, pp. 443-482 | DOI | MR | Zbl

[Bi] S. Billey Kostant polynomials and the cohomology ring for $G / B$ , Duke Math. J, Volume 96 (1999) no. 1, pp. 205-224 | DOI | MR | Zbl

[Bo] A. Borel Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math, Volume 57 (1953), pp. 115-207 | DOI | MR | Zbl

[BS] N. Bergeron; F. Sottile Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J, Volume 95 (1998) no. 2, pp. 373-423 | MR | Zbl

[C-F] I. Ciocan; - Fontanine The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc, Volume 351 (1999) no. 7, pp. 2695-2729 | DOI | MR | Zbl

[D1] M. Demazure Invariants symétriques des groupes de Weyl et torsion, Invent. Math, Volume 21 (1973), pp. 287-301 | DOI | MR | Zbl

[D2] M. Demazure Désingularization des variétés de Schubert généralisées, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), Volume 7 (1974), pp. 53-88 | Numdam | MR | Zbl

[F1] W. Fulton Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J, Volume 65 (1992) no. 3, pp. 381-420 | MR | Zbl

[F2] W. Fulton Schubert varieties in flag bundles for the classical groups, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) (Israel Math. Conf. Proc. 9) (1996), pp. 241-262 | Zbl

[F3] W. Fulton Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom, Volume 43 (1996) no. 2, pp. 276-290 | MR | Zbl

[F4] W. Fulton Intersection Theory, Ergebnisse der Math, 2, Springer-Verlag, Berlin, 1998 | MR | Zbl

[FK] S. Fomin; A. N. Kirillov Combinatorial $B_{n}$ -analogs of Schubert polynomials, Trans. Amer. Math. Soc, Volume 348 (1996) no. 9, pp. 3591-3620 | DOI | MR | Zbl

[FP] W. Fulton; P. Pragacz Schubert varieties and degeneracy loci, Lecture Notes in Math, 1689, Springer-Verlag, Berlin, 1998 | MR | Zbl

[Gra] W. Graham The class of the diagonal in flag bundles, J. Differential Geom, Volume 45 (1997) no. 3, pp. 471-487 | MR | Zbl

[Gro] A. Grothendieck Techniques de construction et théorèmes d'existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki 13 (1960/61) no. 221 | Numdam | Zbl

[HT] J. Harris; L. W. Tu On symmetric and skew-symmetric determinantal varieties, Topology, Volume 23 (1984) no. 1, pp. 71-84 | DOI | MR | Zbl

[JLP] T. Józefiak; A. Lascoux; P. Pragacz Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Math. USSR Izvestija, Volume 18 (1982), pp. 575-586 | DOI | MR | Zbl

[KL] G. Kempf; D. Laksov The determinantal formula of Schubert calculus, Acta Math, Volume 132 (1974), pp. 153-162 | DOI | MR | Zbl

[KT2] A. Kresch; H. Tamvakis Quantum cohomology of orthogonal Grassmannians (2001) (Preprint) | MR

[L] A. Lascoux Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris, Sér. I Math, Volume 295 (1982) no. 5, pp. 393-398 | MR | Zbl

[LP1] A. Lascoux; P. Pragacz Operator calculus for $\tilde{Q}$ -polynomials and Schubert polynomials, Adv. Math, Volume 140 (1998) no. 1, pp. 1-43 | DOI | MR | Zbl

[LP2] A. Lascoux; P. Pragacz Orthogonal divided differences and Schubert polynomials, $\tilde{P}$ -functions, and vertex operators, Michigan Math. J, Volume 48 (2000), pp. 417-441 | DOI | MR | Zbl

[LP3] A. Lascoux; P. Pragacz Schur $Q$ -functions and degeneracy locus formulas for morphisms with symmetries (Recent Progress in Intersection Theory) (2000), pp. 239-263 | Zbl

[LS] A. Lascoux; M.-P. Schützenberger Polynômes de Schubert, C. R. Acad. Sci. Paris, Sér. I Math, Volume 294 (1982) no. 13, pp. 447-450 | MR | Zbl

[M1] I. G. Macdonald Notes on Schubert polynomials, 6, Publ. LACIM, Univ. de Québec à Montréal, Montréal, 1991 | MR

[M2] I. G. Macdonald Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991) (L.M.S. Lecture Note Series), Volume 166 (1991), pp. 73-99 | Zbl

[M3] I. G. Macdonald Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995 | MR | Zbl

[P1] P. Pragacz Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Topics in Algebra, Part 2 (Warsaw, 1988), Volume 26 (1990), pp. 189-199 | Zbl

[P2] P. Pragacz Algebro-geometric applications of Schur $S$ - and $Q$ -polynomials, Séminaire d'Algèbre Dubreil-Malliavin 1989-1990 (Lecture Notes in Math), Volume 1478 (1991), pp. 130-191 | Zbl

[PR1] P. Pragacz; J. Ratajski A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians, J. reine angew. Math, Volume 476 (1996), pp. 143-189 | MR | Zbl

[PR2] P. Pragacz; J. Ratajski Formulas for Lagrangian and orthogonal degeneracy loci; $\tilde{Q}$ -polynomial approach, Compositio Math, Volume 107 (1997) no. 1, pp. 11-87 | DOI | MR | Zbl

[S] I. Schur Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math, Volume 139 (1911), pp. 155-250 | DOI | JFM

[Ta] H. Tamvakis Arakelov theory of the Lagrangian Grassmannian, J. reine angew. Math, Volume 516 (1999), pp. 207-223 | DOI | MR | Zbl

[Tu] L. W. Tu Degeneracy loci, Proc. conf. algebraic geom. (Berlin, 1985) (Teubner-Texte Math), Volume 92 (1986), pp. 296-305 | Zbl

[KT1] A. Kresch; H. Tamvakis Quantum cohomology of the Lagrangian Grassmannian (J. Algebraic Geometry, to appear) | MR | Zbl

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