Skew Schubert functions and the Pieri formula for flag manifolds
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- by Nantel Bergeron and Frank Sottile PDF
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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
Additional Information
- 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
- MR Author ID: 355336
- ORCID: 0000-0003-0087-7120
- Email: sottile@math.umass.edu
- Received by editor(s): October 9, 2000
- Published electronically: 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. - © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 651-673
- MSC (1991): Primary 05E15, 14M15, 05E05, 06A07, 14N10
- DOI: https://doi.org/10.1090/S0002-9947-01-02845-8
- MathSciNet review: 1862562
Dedicated: In memory of Rodica Simion