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Double Schubert polynomials and degeneracy loci for the classical groups. (English) Zbl 1059.14063
The main aim of the present article is to define and study polynomials that the authors propose as type \(B\), \(C\) and \(D\) double Schubert polynomials. For the general linear group the corresponding objects are the double Schubert polynomials of Lascoux and Schützenberger. These type \(A\) polynomials possess a series of remarkable properties and the authors propose a theory with as many of the analogous properties as possible. They succeed in obtaining several properties which are desirable both from the geometric and combinatorial points of view.
When restricted to maximal Grassmannian elements of the Weyl group, the single versions of the polynomials are the \(\widetilde P\)- and \(\widetilde Q\)-polynomials of P. Pragacz and J. Ratajski [J. Reine Angew. Math. 476, 143–189 (1996; Zbl 0847.14029)]. The latter polynomials play, in some sense, the role in types \(B\), \(C\) and \(D\) analogous to that of Schur’s \(S\)-functions in type \(A\). The utility of the \(\widetilde P\)- and \(\widetilde Q\)-polynomials in the description of Schubert calculus and degeneracy loci was studied by P. Pragacz and J. Ratajski [Compos. Math. 107, No. 1, 11–87 (1997; Zbl 0916.14026)], and according to the authors [see Compos. Math. 140, No. 2, 482–500 (2004; Zbl 1077.14083) and J. Reine Angew. Math. 516, 207–223 (1999; Zbl 0934.14018)] the multiplication of \(\widetilde Q\)-polynomials describes both the arithmetic and quantum Schubert calculus on the Lagrangian Grassmannian. Thus the double Schubert polynomials in the present article are closely related to natural families of representing polynomials.
In many cases the authors obtain an analogue of the determinantal formula for Schubert cycles in Grassmannians and they answer a question of W. Fulton and P. Pragacz [Schubert varieties and degeneracy loci. Lect. Notes Math. 1689 (1998; Zbl 0913.14016)]. The formulas generalize those obtained by Pragacz and Ratajski [loc. cit.].
The main ingredients in the proofs are the geometric work of W. Fulton [Duke Math. J. 65, No. 3, 381–420 (1992; Zbl 0788.14044) and Isr. Math. Conf. Proc. 9, 241–262 (1996; Zbl 0862.14032)] and W. Graham [J. Differ. Geom. 45, 471–487 (1997; Zbl 0935.14015)] and the algebraic tools developed by A. Lascoux and P. Pragacz [Adv. Math. 140, No. 1, 1–43 (1998; Zbl 0951.14035) and Mich. Math. J. 48, Spec. Vol., 417–441 (2000; Zbl 1003.05106)].

MSC:
14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
05E15 Combinatorial aspects of groups and algebras (MSC2010)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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References:
[1] An algebraic formula for the Gysin homomorphism from \(G/B\) to \(G/P,\) Illinois J. Math, 31, 2, 312-320, (1987) · Zbl 0629.57030
[2] Quantum Schubert calculus, Adv. Math, 128, 2, 289-305, (1997) · Zbl 0945.14031
[3] Schubert cells and cohomology of the spaces \(G/P,\) Russian Math. Surveys, 28, 3, 1-26, (1973) · Zbl 0289.57024
[4] Schubert polynomials for the classical groups, J. Amer. Math. Soc, 8, 2, 443-482, (1995) · Zbl 0832.05098
[5] Kostant polynomials and the cohomology ring for \(G/B,\) Duke Math. J, 96, 1, 205-224, (1999) · Zbl 0980.22018
[6] Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math, 57, 115-207, (1953) · Zbl 0052.40001
[7] Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J, 95, 2, 373-423, (1998) · Zbl 0939.05084
[8] The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc, 351, 7, 2695-2729, (1999) · Zbl 0920.14027
[9] Invariants symétriques des groupes de Weyl et torsion, Invent. Math, 21, 287-301, (1973) · Zbl 0269.22010
[10] Désingularization des variétés de Schubert généralisées, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 7, 53-88, (1974) · Zbl 0312.14009
[11] Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J, 65, 3, 381-420, (1992) · Zbl 0788.14044
[12] Schubert varieties in flag bundles for the classical groups, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 241-262, (1996) · Zbl 0862.14032
[13] Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom, 43, 2, 276-290, (1996) · Zbl 0911.14001
[14] Intersection Theory, 2, (1998), Springer-Verlag, Berlin · Zbl 0885.14002
[15] Combinatorial \(B_n\)-analogs of Schubert polynomials, Trans. Amer. Math. Soc, 348, 9, 3591-3620, (1996) · Zbl 0871.05060
[16] Schubert varieties and degeneracy loci, 1689, (1998), Springer-Verlag, Berlin · Zbl 0913.14016
[17] The class of the diagonal in flag bundles, J. Differential Geom, 45, 3, 471-487, (1997) · Zbl 0935.14015
[18] Techniques de construction et théorèmes d’existence en géométrie algébrique IV: LES schémas de Hilbert, Séminaire Bourbaki 13, 221, (196061) · Zbl 0236.14003
[19] On symmetric and skew-symmetric determinantal varieties, Topology, 23, 1, 71-84, (1984) · Zbl 0534.55010
[20] Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Math. USSR Izvestija, 18, 575-586, (1982) · Zbl 0489.14020
[21] The determinantal formula of Schubert calculus, Acta Math, 132, 153-162, (1974) · Zbl 0295.14023
[22] Quantum cohomology of orthogonal Grassmannians, (2001)
[23] Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris, Sér. I Math, 295, 5, 393-398, (1982) · Zbl 0495.14032
[24] Operator calculus for \(\widetilde Q\)-polynomials and Schubert polynomials, Adv. Math, 140, 1, 1-43, (1998) · Zbl 0951.14035
[25] Orthogonal divided differences and Schubert polynomials \(, \widetilde P\)-functions, and vertex operators, Michigan Math. J, 48, 417-441, (2000) · Zbl 1003.05106
[26] Schur \(Q\)-functions and degeneracy locus formulas for morphisms with symmetries, 239-263, (2000), Birkhäuser, Boston · Zbl 0969.14033
[27] Polynômes de Schubert, C. R. Acad. Sci. Paris, Sér. I Math, 294, 13, 447-450, (1982) · Zbl 0495.14031
[28] Notes on Schubert polynomials, 6, (1991), Publ. LACIM, Univ. de Québec à Montréal, Montréal
[29] Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991), 166, 73-99, (1991), Cambridge Univ. Press, Cambridge · Zbl 0784.05061
[30] Symmetric Functions and Hall Polynomials, (1995), Clarendon Press, Oxford · Zbl 0824.05059
[31] Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Topics in Algebra, Part 2 (Warsaw, 1988), 26, 189-199, (1990), Banach Center Publ, Part 2, PWN, Warsaw · Zbl 0743.14009
[32] Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials, Séminaire d’Algèbre Dubreil-Malliavin 1989-1990, 1478, 130-191, (1991), Springer-Verlag, Berlin · Zbl 0783.14031
[33] A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians, J. reine angew. Math, 476, 143-189, (1996) · Zbl 0847.14029
[34] Formulas for Lagrangian and orthogonal degeneracy loci;\( \widetilde Q\)-polynomial approach, Compositio Math, 107, 1, 11-87, (1997) · Zbl 0916.14026
[35] Über die darstellung der symmetrischen und der alternierenden gruppe durch gebrochene lineare substitutionen, J. reine angew. Math, 139, 155-250, (1911) · JFM 42.0154.02
[36] Arakelov theory of the Lagrangian Grassmannian, J. reine angew. Math, 516, 207-223, (1999) · Zbl 0934.14018
[37] Degeneracy loci, Proc. conf. algebraic geom. (Berlin, 1985), 92, 296-305, (1986), Teubner, Leipzig · Zbl 0626.14019
[38] Quantum cohomology of the Lagrangian Grassmannian · Zbl 1051.53070
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