Skew Schubert polynomials

Lenart, Cristian; Sottile, Frank

doi:10.1090/S0002-9939-03-06919-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

Skew Schubert polynomials

Authors: Cristian Lenart and Frank Sottile
Journal: Proc. Amer. Math. Soc. 131 (2003), 3319-3328
MSC (2000): Primary 05E05, 14M15, 06A07
Posted: February 20, 2003
MathSciNet review: 1990619
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.

1. Nantel Bergeron and Sara Billey, RC-graphs and Schubert polynomials, Experiment. Math. 2 (1993), no. 4, 257–269. MR 1281474 (95g:05107)
2. Nantel Bergeron and Frank Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), no. 2, 373–423. MR 1652021 (2000d:05127), http://dx.doi.org/10.1215/S0012-7094-98-09511-4
3. -, A Pieri-type formula for isotropic flag manifolds, 2002, Trans. Amer. Math. Soc., 354 No. 7, (2002), 2659-2705.
4. Nantel Bergeron and Frank Sottile, Skew Schubert functions and the Pieri formula for flag manifolds, Trans. Amer. Math. Soc. 354 (2002), no. 2, 651–673 (electronic). MR 1862562 (2002k:05243), http://dx.doi.org/10.1090/S0002-9947-01-02845-8
5. Sara C. Billey, William Jockusch, and Richard P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345–374. MR 1241505 (94m:05197), http://dx.doi.org/10.1023/A:1022419800503
6. Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565–596. MR 1431829 (98d:14063), http://dx.doi.org/10.1090/S0894-0347-97-00237-3
7. Sergey Fomin and Anatol N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), 1996, pp. 123–143. MR 1394950 (98b:05101), http://dx.doi.org/10.1016/0012-365X(95)00132-G
8. Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793 (95f:05115), http://dx.doi.org/10.1006/aima.1994.1009
9. William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693 (99f:05119)
10. Anatol N. Kirillov, Skew divided difference operators and Schubert polynomials, www.arXiv.org/math.QA/9712053.
11. 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 (2001f:05161), http://dx.doi.org/10.1016/S0012-365X(99)00263-0
12. Allen Knutson and Ezra Miller, Gröbner geometry of Schubert polynomials, www.arXiv.org/math.AG/0110058.
13. M. Kogan and A. Kumar, A proof of Pieri's formula using generalized Schensted insertion algorithm for rc-graphs, Proc. Amer. Math. Soc. 130 (2002), 2525-2534.
14. 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 (83e:14039), http://dx.doi.org/10.1090/conm/088/1000001
15. I.G. Macdonald, Notes on Schubert polynomials, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Univ. du Québec à Montréal, Montréal, 1991.
16. Laurent Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spécialisés, no. 3, Soc. Math. France, 1998.
17. Ezra Miller, Mitosis recursion for coefficients of Schubert polynomials, to appear in J. Combin. Theory Ser. A.
18. Frank Sottile, Pieri’s formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89–110 (English, with English and French summaries). MR 1385512 (97g:14035)
19. Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Springer, Berlin, 1977, pp. 217–251. Lecture Notes in Math., Vol. 579. MR 0465880 (57 #5766)
20. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282 (2000k:05026)
21. Richard P. Stanley, Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319. MR 1754784 (2001f:05001)
22. Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|