Skew Schubert functions and the Pieri formula for flag manifolds

Bergeron, Nantel; Sottile, Frank

doi:10.1090/S0002-9947-01-02845-8

Skew Schubert functions and the Pieri formula for flag manifolds
HTML articles powered by AMS MathViewer

by Nantel Bergeron and Frank Sottile PDF

Trans. Amer. Math. Soc. 354 (2002), 651-673 Request permission

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.

References

A. D. Berenshteĭn and A. V. Zelevinskiĭ, Involutions on Gel′fand-Tsetlin schemes and multiplicities in skew $\textrm {GL}_n$-modules, Dokl. Akad. Nauk SSSR 300 (1988), no. 6, 1291–1294 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 799–802. MR 950493
Nantel Bergeron, A combinatorial construction of the Schubert polynomials, J. Combin. Theory Ser. A 60 (1992), no. 2, 168–182. MR 1168152, DOI 10.1016/0097-3165(92)90002-C
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, DOI 10.1215/S0012-7094-98-09511-4
Nantel Bergeron and Frank Sottile, A monoid for the Grassmannian Bruhat order, European J. Combin. 20 (1999), no. 3, 197–211. MR 1687251, DOI 10.1006/eujc.1999.0283
I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Schubert cells and cohomology of the spaces $G/P$, Russian Mathematical Surveys, 28 (1973), pp. 1–26.
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, DOI 10.1023/A:1022419800503
C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1–23 (French). With a foreword by Armand Borel. MR 1278698
Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697
Paul Edelman and Curtis Greene, Balanced tableaux, Adv. in Math. 63 (1987), no. 1, 42–99. MR 871081, DOI 10.1016/0001-8708(87)90063-6
Richard Ehrenborg and Margaret A. Readdy, Sheffer posets and $r$-signed permutations, Ann. Sci. Math. Québec 19 (1995), no. 2, 173–196 (English, with English and French summaries). MR 1365825
Sergey Fomin and Anatol N. Kirillov, Combinatorial $B_n$-analogues of Schubert polynomials, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3591–3620. MR 1340174, DOI 10.1090/S0002-9947-96-01558-9
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, DOI 10.1016/0012-365X(95)00132-G
Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793, DOI 10.1006/aima.1994.1009
Mark D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), no. 1-3, 79–113. MR 1158783, DOI 10.1016/0012-365X(92)90368-P
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.
Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727. MR 272654
Axel Kohnert, Weintrauben, Polynome, Tableaux, Bayreuth. Math. Schr. 38 (1991), 1–97 (German). Dissertation, Universität Bayreuth, Bayreuth, 1990. MR 1132534
Witold Kraśkiewicz, Reduced decompositions in Weyl groups, European J. Combin. 16 (1995), no. 3, 293–313. MR 1330543, DOI 10.1016/0195-6698(95)90033-0
Witold Kraśkiewicz and Piotr Pragacz, Foncteurs de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 9, 209–211 (French, with English summary). MR 883476
G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333–350 (French). MR 309747, DOI 10.1016/0012-365X(72)90041-6
Alain Lascoux and Marcel-P. Schützenberger, Le monoïde plaxique, Noncommutative structures in algebra and geometric combinatorics (Naples, 1978) Quad. “Ricerca Sci.”, vol. 109, CNR, Rome, 1981, pp. 129–156 (French, with Italian summary). MR 646486
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
Alain Lascoux and Marcel-Paul Schützenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 11, 629–633 (French, with English summary). MR 686357
Alain Lascoux and Marcel-Paul Schützenberger, Symmetry and flag manifolds, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 118–144. MR 718129, DOI 10.1007/BFb0063238
I. G. Macdonald, Notes on Schubert Polynomials, Laboratoire de combinatoire et d’informatique mathématique (LACIM), Université du Québec à Montréal, Montréal, 1991.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Laurent Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spécialisés [Specialized Courses], vol. 3, Société Mathématique de France, Paris, 1998 (French, with English and French summaries). MR 1638048
D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253–286. MR 106911, DOI 10.1112/plms/s3-9.2.253
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
Jeffrey B. Remmel and Mark Shimozono, A simple proof of the Littlewood-Richardson rule and applications, Discrete Math. 193 (1998), no. 1-3, 257–266. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661373, DOI 10.1016/S0012-365X(98)00145-9
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
Richard P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), no. 4, 359–372. MR 782057, DOI 10.1016/S0195-6698(84)80039-6
S. Veigneau, Calcul Symbolique et Calcul Distribué en Combinatoire Algebrique, Thèse, l’Université de Marne-la-Valeé, 1996.
Rudolf Winkel, Diagram rules for the generation of Schubert polynomials, J. Combin. Theory Ser. A 86 (1999), no. 1, 14–48. MR 1682961, DOI 10.1006/jcta.1998.2931
Rudolf Winkel, On the multiplication of Schubert polynomials, Adv. in Appl. Math. 20 (1998), no. 1, 73–97. MR 1488233, DOI 10.1006/aama.1997.0566