Transactions of the American Mathematical Society

Author(s): Nantel Bergeron; Frank Sottile
Journal: Trans. Amer. Math. Soc. 354 (2002), 2659-2705.
MSC (2000): Primary 14M15, 05E15, 05E05, 06A07, 14N10
Posted: February 20, 2002
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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

(respectively, type

) Schubert polynomial by the Schur

-polynomial

(respectively, the Schur

-polynomial

). 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.

1.: N. BERGERON AND S. BILLEY, RC-Graphs and Schubert polynomials, Experimental Math., 2 (1993), pp. 257-269. MR 95g:05107
2.: 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. CMP 2001:01
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.: -, Hopf algebras of edge-labeled posets, J. Algebra, 216 (1999), pp. 641-651. MR 2000e:16033
5.: -, A monoid for the Grassmannian Bruhat order, European J. Combin., 20 (1999), pp. 197-211. MR 2000f:05091
6.: 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.
7.: S. BILLEY AND M. HAIMAN, Schubert polynomials for the classical groups, J. AMS, 8 (1995), pp. 443-482. MR 98e:05109
8.: B. BOE AND H. HILLER, Pieri formula for ${S}{O}_{2n+1}/{U}_n$ and ${S}{P}_n/{U}_n$ , Adv. in Math., 62 (1986), pp. 49-67. MR 87k:14058
9.: A. BOREL, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts, Ann. Math., 57 (1953), pp. 115-207. MR 14:490e
10.: C. CHEVALLEY, Sur les décompositions cellulaires des espaces , in Algebraic Groups and their Generalizations: Classical Methods, W. Haboush, ed., vol. 56, Part 1 of Proc. Sympos. Pure Math., Amer. Math. Soc., 1994, pp. 1-23. MR 95e:14041
11.: M. DEMAZURE, Désingularization des variétés de Schubert généralisées, Ann. Sc. E. N. S. (4), 7 (1974), pp. 53-88. MR 50:7174
12.: V. DEODHAR, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math., 79 (1985), pp. 499-511. MR 86f:20045
13.: D. EDIDIN AND W. GRAHAM, Characteristic classes and quadric bundles, Duke Math. J., 78 (1995), pp. 277-299. MR 98a:14004
14.: S. FOMIN AND A. N. KIRILLOV, Combinatorial ${B}_n$ -analogs of Schubert polynomials, Trans. AMS, 348 (1996), pp. 3591-3620. MR 98e:05110
15.: W. FULTON, Intersection Theory, no. 2 in Ergebnisse der Math., Springer-Verlag, 1984. MR 85k:14004
16.: -, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Diff. Geom., 43 (1996), pp. 276-290. MR 98d:14004
17.: W. FULTON AND P. PRAGACZ, Schubert Varieties and Degeneracy Loci, Lec. Notes in Math. 1689, Springer-Verlag, 1998. MR 99m:14092
18.: T. JÓZEFIAK, Schur -functions and the cohomology of isotropic Grassmannians, Math. Proc. Camb. Phil. Soc., 109 (1991), pp. 471-478. MR 92a:57046
19.: S. KLEIMAN, The transversality of a general translate, Comp. Math., 28 (1974), pp. 287-297. MR 50:13063
20.: M. KOGAN AND A. KUMAR, A proof of Pieri's formula using generalized Schensted insertion algorithm for -graphs, arXiv.math.CO/0010109.
21.: G. KREWERAS, Sur les partitions non croisees d'un cycle, Discrete Math., 1 (1972), pp. 333-350. MR 46:8852
22.: A. LASCOUX AND M.-P. SCHÜTZENBERGER, Polynômes de Schubert, C. R. Acad. Sci. Paris, 294 (1982), pp. 447-450. MR 83e:14039
23.: -, 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
24.: I. G. MACDONALD, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, 1995.
second edition. MR 96h:05207
25.: A. POSTNIKOV, On a quantum version of Pieri's formula.
Advances in Geometry, J.-L. Brylinski and R. Brylinski, eds., Progr. Math. 172, Birkhäuser, 1999. 371-383. MR 99m:14096
26.: A. POSTNIKOV, Symmetries of Gromov-Witten invariants.
Advances in Algebraic Geometry Motivated by Physics, E.Previato, ed., Contemp. Math., 276, AMS, 2001. 251-258.
27.: P. PRAGACZ, Algebro-geometric applications of Schur -and -polynomials, in Topics in Invariant Theory, Séminaire d'Algèbre Dubreil-Malliavin 1989-90, Springer-Verlag, 1991, pp. 130-191. MR 93h:05170
28.: P. PRAGACZ AND J. RATAJSKI, Pieri-type formula for and , C. R. Acad. Sci. Paris, 317 (1993), pp. 1035-1040. MR 94i:14055
29.: -, Pieri-type formula for Lagrangian and odd orthogonal Grassmannians, J. reine agnew. Math., 476 (1996), pp. 143-189. MR 97i:14032
30.: -, A Pieri-type theorem for even orthogonal Grassmannians.
Max-Planck Institut preprint, 1996.
31.: -, Formulas for Lagrangian and orthogonal loci: The $\tilde{Q}$ -polynomial approach, Composito Math., 107 (1997), pp. 11-87. MR 98g:14063
32.: F. SOTTILE, Pieri's formula for flag manifolds and Schubert polynomials, Annales de l'Institut Fourier, 46 (1996), pp. 89-110. MR 97g:14035
33.: -, Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloq. Math., 82, (1999), pp. 49-63. MR 2001a:14053
34.: J. STEMBRIDGE, Shifted tableaux and the projective representations of the symmetric group, Adv. Math., 74 (1989), pp. 87-134. MR 90k:20026
35.: -, Enriched -partitions, Trans. AMS, 349 (1997), pp. 763-788. MR 97f:06006

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M15, 05E15, 05E05, 06A07, 14N10

Nantel Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, 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/S0002-9947-02-02946-X
PII: S 0002-9947(02)02946-X
Received by editor(s): March 7, 2001
Received by editor(s) in revised form: August 6, 2001
Posted: February 20, 2002
Additional Notes: First author supported in part by NSERC and CRM grants.
Second author supported in part by NSERC grant OGP0170279 and NSF grant DMS-9022140.
Copyright of article: Copyright 2002, American Mathematical Society

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