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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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