Quantum Bruhat graph and Schubert polynomials

Postnikov, Alexander

doi:10.1090/S0002-9939-04-07614-2

Quantum Bruhat graph and Schubert polynomials
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by Alexander Postnikov PDF

Proc. Amer. Math. Soc. 133 (2005), 699-709 Request permission

Abstract:

The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of $G/B$. We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph. We define path Schubert polynomials, which are quantum cohomology analogs of skew Schubert polynomials recently introduced by Lenart and Sottile. They are given by sums over paths in the quantum Bruhat graph of type $A$. The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes.

References

Francesco Brenti, Sergey Fomin, and Alexander Postnikov, Mixed Bruhat operators and Yang-Baxter equations for Weyl groups, Internat. Math. Res. Notices 8 (1999), 419–441. MR 1687323, DOI 10.1155/S1073792899000215
Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485–524. MR 1695799, DOI 10.1215/S0012-7094-99-09815-0
Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565–596. MR 1431829, DOI 10.1090/S0894-0347-97-00237-3
Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182. MR 1667680
William Fulton, Christopher Woodward: On the quantum product of Schubert classes, e-print ArXiv arXiv:math.AG/0112183.
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
Anatol N. Kirillov and Toshiaki Maeno, Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, Discrete Math. 217 (2000), no. 1-3, 191–223 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766267, DOI 10.1016/S0012-365X(99)00263-0
Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
Cristian Lenart, Frank Sottile: Skew Schubert polynomials, Proceedings of the American Mathematical Society 131 (2003), no. 11, 3319–3328.
I. G. Macdonald, Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991) London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 73–99. MR 1161461
Dale Peterson: Lectures on quantum cohomology of $G/P$, M.I.T., 1996.
Alexander Postnikov, On a quantum version of Pieri’s formula, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 371–383. MR 1667687
Alexander Postnikov, Symmetries of Gromov-Witten invariants, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 251–258. MR 1837121, DOI 10.1090/conm/276/04524
Alexander Postnikov: Affine approach to quantum Schubert calculus, e-print arXiv:math.CO/0205165.