A Pieri-type formula for isotropic flag manifolds

Bergeron, Nantel; Sottile, Frank

doi:10.1090/S0002-9947-02-02946-X

A Pieri-type formula for isotropic flag manifolds
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by Nantel Bergeron and Frank Sottile PDF

Trans. Amer. Math. Soc. 354 (2002), 2659-2705 Request permission

Abstract:

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants.

References

Nantel Bergeron and Sara Billey, RC-graphs and Schubert polynomials, Experiment. Math. 2 (1993), no. 4, 257–269. MR 1281474
N. Bergeron, S. Mykytiuk, F. Sottile, and S. van Willigenburg, Non-commutative Pieri operations on posets, J. Combin. Th. Ser. A, 91 (2000), 84–110.
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, Hopf algebras and edge-labeled posets, J. Algebra 216 (1999), no. 2, 641–651. MR 1692973, DOI 10.1006/jabr.1998.7794
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 Billey and Mark Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), no. 2, 443–482. MR 1290232, DOI 10.1090/S0894-0347-1995-1290232-1
Howard Hiller and Brian Boe, Pieri formula for $\textrm {SO}_{2n+1}/\textrm {U}_n$ and $\textrm {Sp}_n/\textrm {U}_n$, Adv. in Math. 62 (1986), no. 1, 49–67. MR 859253, DOI 10.1016/0001-8708(86)90087-3
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
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
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
Dan Edidin and William Graham, Characteristic classes and quadric bundles, Duke Math. J. 78 (1995), no. 2, 277–299. MR 1333501, DOI 10.1215/S0012-7094-95-07812-0
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
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
William Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom. 43 (1996), no. 2, 276–290. MR 1424427
William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998. Appendix J by the authors in collaboration with I. Ciocan-Fontanine. MR 1639468, DOI 10.1007/BFb0096380
Tadeusz Józefiak, Schur $Q$-functions and cohomology of isotropic Grassmannians, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 3, 471–478. MR 1094746, DOI 10.1017/S0305004100069917
Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
M. Kogan and A. Kumar, A proof of Pieri’s formula using generalized Schensted insertion algorithm for $RC$-graphs, arXiv.math.CO/0010109.
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-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, 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, 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
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
A. Postnikov, Symmetries of Gromov-Witten invariants. Advances in Algebraic Geometry Motivated by Physics, E.Previato, ed., Contemp. Math., 276, AMS, 2001. 251–258.
Piotr Pragacz, Algebro-geometric applications of Schur $S$- and $Q$-polynomials, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130–191. MR 1180989, DOI 10.1007/BFb0083503
Piotr Pragacz and Jan Ratajski, A Pieri-type formula for $\textrm {Sp}(2m)/P$ and $\textrm {SO}(2m+1)/P$, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 11, 1035–1040 (English, with English and French summaries). MR 1249784
Piotr Pragacz and Jan Ratajski, A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians, J. Reine Angew. Math. 476 (1996), 143–189. MR 1401699
—, A Pieri-type theorem for even orthogonal Grassmannians. Max-Planck Institut preprint, 1996.
P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; $\~Q$-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87. MR 1457343, DOI 10.1023/A:1000182205320
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
Frank Sottile, Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloq. Math. 82 (1999), no. 1, 49–63. MR 1736034, DOI 10.4064/cm-82-1-49-63
John R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), no. 1, 87–134. MR 991411, DOI 10.1016/0001-8708(89)90005-4
John R. Stembridge, Enriched $P$-partitions, Trans. Amer. Math. Soc. 349 (1997), no. 2, 763–788. MR 1389788, DOI 10.1090/S0002-9947-97-01804-7