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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:
 [1] S. Abeasis, A. Del Fra, and H. Kraft, The geometry of representations of $$A\sbm$$ , Math. Ann. 256 (1981), no. 3, 401-418. · Zbl 0477.14027 [2] I. N. Bernšteĭ n, I. M. Gel’fand, and S. I. Gel’fand, Schubert cells, and the cohomology of the spaces $$G/P$$ , Uspehi Mat. Nauk 28 (1973), no. 3(171), 3-26. · Zbl 0289.57024 [3] A. Buch and W. Fulton, Chern class formulas for quiver varieties , to appear in Invent. Math. · Zbl 0942.14027 [4] Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties , Internat. Math. Res. Notices (1995), no. 6, 263-277. · Zbl 0847.14011 [5] I. Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties , · Zbl 0969.14039 [6] I. Ciocan-Fontanine and W. Fulton, Quantum double Schubert polynomials 6, Institut Mittag-Leffler, 1996/97, Appendix J in [FP]. [7] Michel Demazure, Désingularisation des variétés de Schubert généralisées , Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. · Zbl 0312.14009 [8] Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials , J. Amer. Math. Soc. 10 (1997), no. 3, 565-596, 2d ed. JSTOR: · Zbl 0912.14018 [9] William Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. · Zbl 0885.14002 [10] William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas , Duke Math. J. 65 (1992), no. 3, 381-420. · Zbl 0788.14044 [11] William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci , Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998. · Zbl 0913.14016 [12] A. N. Kirillov, Quantum Schubert polynomials and quantum Schur functions , · Zbl 1040.05029 [13] A. N. Kirillov and T. Maeno, Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula , · Zbl 0958.05137 [14] V. Lakshmibai and Peter Magyar, Degeneracy schemes, quiver schemes, and Schubert varieties , Internat. Math. Res. Notices (1998), no. 12, 627-640. · Zbl 0936.14001 [15] Alain Lascoux, Classes de Chern des variétés de drapeaux , C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 5, 393-398. · Zbl 0495.14032 [16] Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert , C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447-450. · Zbl 0495.14031 [17] A. Lascoux and M.-P. Schützenberger, Fonctorialité des polynômes de Schubert , Invariant theory (Denton, TX, 1986), Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 585-598. · Zbl 0697.20003 [18] I. G. Macdonald, Notes on Schubert polynomials , Monographies du LaCIM, vol. 6, Université du Québec, Montréal, 1991. · Zbl 0784.05061 [19] Frank Sottile, Pieri’s formula for flag manifolds and Schubert polynomials , Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89-110. · Zbl 0837.14041
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