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Universal Schubert polynomials. (English) Zbl 0981.14022
In this fundamental and beautiful article the author introduces universal Schubert polynomials that specialize to all previously known Schubert polynomials: those of Lascoux and Schützenberger, the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov, and the quantum Schubert polynomials for partial flag varieties of Ciocan-Fontanine. Also double versions of these polynomials are given, that generalize the previously known double Schubert polynomials of Lascoux, MacDonald, Kirillov and Maeno, and those of Ciocan-Fontanine and Fulton.
The universal Schubert polynomials describe degeneracy loci of maps of vector bundles, in a more general setting than that of the author’s beautiful earlier article [W. Fulton, Duke Math. J. 65, 381-420 (1992; Zbl 0788.14044)].
The setting is a sequence of maps of locally free \(\mathcal O_X\)-modules \[ F_1\to F_2\to \cdots \to F_n \to E_n \to \cdots \to E_2\to E_1 \] on a scheme \(X\). In contrast to the mentioned article (loc. cit.) the maps \(F_i \to F_{i+1}\) do not have to be injective and the maps \(E_{i+1} \to E_i\) do not have to be surjective. For each \(w\) in the symmetric group \(S_{n+1}\), there is a degeneracy locus \[ \Omega_w =\{x\in X\mid \text{rank}(F_q(x) \to E_p(x)) \leq r_w(p,q) \text{ for all } 1\leq p, q\leq n\}, \] where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i)\leq q\). Such degeneracy loci are described by the double form \({\mathfrak S}_w(c,d)\) of universal Schubert polynomials evaluated at the Chern classes of all the bundles involved.
The classical approaches of Demazure, or Bernstein, Gel’fand, and Gel’fand do not work in this case. Instead a locus in a flag bundle is found that maps to a given degeneracy locus \(\Omega_w\), such that one has injections and surjections of the bundles involved, and such that the results of the article mentioned above can be applied. Then the formula is pushed forward to get a formula for \(\Omega_w\).

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
05E15 Combinatorial aspects of groups and algebras (MSC2010)
14M12 Determinantal varieties
57R20 Characteristic classes and numbers in differential topology
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References:
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