Transactions of the American Mathematical Society

Author(s): Nantel Bergeron; Frank Sottile
Journal: Trans. Amer. Math. Soc. 354 (2002), 651-673.
MSC (1991): Primary 05E15, 14M15, 05E05, 06A07, 14N10
Posted: September 21, 2001
Retrieve article in: PDF
This article is available free of charge

Abstract: We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.

1.: A. D. BERENSHTEN AND A. V. ZELEVINSKI, Involutions on Gelfand-Tsetlin schemes and multiplicities in skew ${\rm {G}{L}}\sb n$ -modules, Dokl. Akad. Nauk SSSR, 300 (1988), pp. 1291-1294. MR 89h:22030
2.: N. BERGERON, A combinatorial construction of the Schubert polynomials, J. Combin. Theory, Ser. A, 60 (1992), pp. 168-182. MR 93i:05138
3.: N. BERGERON AND F. SOTTILE, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds.
Duke Math. J., 95 (1998), pp. 373-423. MR 2000d:05127
4.: -, A monoid for the Grassmannian Bruhat order. Europ. J. Combinatorics, 20 (1999), pp. 197-211. MR 2000f:05091
5.: I. N. BERNSTEIN, I. M. GELFAND, AND S. I. GELFAND, Schubert cells and cohomology of the spaces , Russian Mathematical Surveys, 28 (1973), pp. 1-26.
6.: S. BILLEY, W. JOCKUSCH, AND R. STANLEY, Some combinatorial properties of Schubert polynomials, J. Algebraic Combinatorics, 2 (1993), pp. 345-374. MR 94m:05197
7.: C. CHEVALLEY, Sur les décompositions cellulaires des espaces , in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1-23.
Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1. MR 95e:14041
8.: M. DEMAZURE, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4), 7 (1974), pp. 53-88. MR 50:7174
9.: P. EDELMAN AND C. GREENE, Balanced tableaux, Adv. Math., 63 (1987), pp. 42-99. MR 88b:05012
10.: R. EHRENBORG AND M. READDY, Sheffer posets and -signed permutations, Ann. Sci. Math Québec, 19 (1995), pp. 173-196. MR 96m:06005
11.: S. FOMIN AND A. N. KIRILLOV, Combinatorial ${B}_n$ -analogs of Schubert polynomials, Trans. Amer. Math. Soc., 348 (1996), pp. 3591-3620. MR 98e:05110
12.: S. FOMIN AND A. N. KIRILLOV, Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete Math., 153 (1996), pp. 124-143.
Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). MR 98b:05101
13.: S. FOMIN AND R. STANLEY, Schubert polynomials and the nil-Coxeter algebra, Adv. Math., 103 (1994), pp. 196-207. MR 95f:05115
14.: M. HAIMAN, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math., 99 (1992), pp. 79-113. MR 93h:05173
15.: A. KIRILLOV AND T. MAENO, Quantum double Schubert polynomials, quantum Schubert polynomials, and the Vafa-Intriligator formula.
Formal power series and algebraic combinatorics (Vienna, 1997). Discrete Math. 217 (2000), no. 1-3, pp. 191-223. CMP 2000:15
16.: D. KNUTH, Permutations, matrices and generalized Young tableaux, Pacific J. Math., 34 (1970), pp. 709-727. MR 42:7535
17.: A. KOHNERT, Weintrauben, Polynome, Tableaux, Bayreuth Math. Schrift., 38 (1990), pp. 1-97. MR 93d:05155
18.: W. KRÁSKIEWICZ, Reduced decompositions in Weyl groups, Europ. J. Combin., 16 (1995), pp. 293-313. MR 96i:05181
19.: W. KRÁSKIEWICZ AND P. PRAGACZ, Foncteurs de Schubert, C. R. Acad. Sci. Paris, 304 (1987), pp. 209-211. MR 89b:14068
20.: G. KREWERAS, Sur les partitions non croisées d'un cycle, Discrete Math., 1 (1972), pp. 333-350. MR 46:8852
21.: A. LASCOUX AND M.-P. SCHÜTZENBERGER, Le monoïde plaxique, in Non-Commutative Structures in Algebra and Geometric Combinatorics, Quad. ``Ricera Sci.,'' 109 Roma, CNR, 1981, pp. 129-156. MR 83g:20016
22.: -, Polynômes de Schubert, C. R. Acad. Sci. Paris, 294 (1982), pp. 447-450. MR 83e:14039
23.: -, Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris, 295 (1982), pp. 629-633. MR 84b:14030
24.: -, Symmetry and flag manifolds, in Invariant Theory, (Montecatini, 1982), vol. 996 of Lecture Notes in Math., Springer-Verlag, 1983, pp. 118-144. MR 85e:14073
25.: I. G. MACDONALD, Notes on Schubert Polynomials, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Université du Québec à Montréal, Montréal, 1991.
26.: -, Symmetric Functions and Hall Polynomials, Oxford University Press, 1995.
second edition. MR 96h:05207
27.: L. MANIVEL, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Société Mathématique de France, Paris, 1998, 179 pages. MR 99k:05159
28.: D. MONK, The geometry of flag manifolds, Proc. London Math. Soc., 9 (1959), pp. 253-286. MR 21:5641
29.: A. POSTNIKOV, On a quantum version of Pieri's formula.
Advances in geometry, Progr. Math., 172, Birkhäuser Boston, Boston, MA (1999) pp. 371-383. MR 99m:14096
30.: J. B. REMMEL AND M. SHIMOZONO, A simple proof of the Littlewood-Richardson rule and applications, Discrete Math., 193 (1998), pp. 257-266.
Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 2000a:05213
31.: F. SOTTILE, Pieri's formula for flag manifolds and Schubert polynomials, Annales de l'Institut Fourier, 46 (1996), pp. 89-110. MR 97g:14035
32.: R. STANLEY, On the number of reduced decompositions of elements of Coxeter groups, Europ. J. Combin., 5 (1984), pp. 359-372. MR 86i:05011
33.: S. VEIGNEAU, Calcul Symbolique et Calcul Distribué en Combinatoire Algebrique, Thèse, l'Université de Marne-la-Valeé, 1996.
34.: R. WINKEL, Diagram rules for the generation of Schubert polynomials.
J. Combin. Theory Ser. A 86 (1999), no. 1, pp 14-48. MR 2001b:05227
35.: -, On the multiplication of Schubert polynomials.
Adv. in Appl. Math. 20 (1998), no. 1, pp 73-97. MR 99c:05200

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 05E15, 14M15, 05E05, 06A07, 14N10

Nantel Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P, Canada
Email: bergeron@mathstat.yorku.ca

Frank Sottile
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
Email: sottile@math.umass.edu

DOI: 10.1090/S0002-9947-01-02845-8
PII: S 0002-9947(01)02845-8
Keywords: Pieri formula, Bruhat order, Schubert polynomial, Stanley symmetric function, flag manifold, {\em jeu de taquin}, weak order
Received by editor(s): October 9, 2000
Posted: September 21, 2001
Additional Notes: The first author was supported in part by NSERC and CRM grants.
The second author was supported in part by NSERC grant OGP0170279 and NSF grant DMS-9022140.
Dedicated: In memory of Rodica Simion
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS