Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A Pieri-type formula for isotropic flag manifolds


Authors: Nantel Bergeron and Frank Sottile
Journal: Trans. Amer. Math. Soc. 354 (2002), 2659-2705
MSC (2000): Primary 14M15, 05E15, 05E05, 06A07, 14N10
DOI: https://doi.org/10.1090/S0002-9947-02-02946-X
Published electronically: February 20, 2002
MathSciNet review: 1895198
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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 $G/P$, 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 $G/B$, 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 $Q$-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 $RC$-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 ${S}$-and ${Q}$-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 ${S}{P}(2m)/{P}$ and ${S}{O}(2m+1)/{P}$, 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 ${P}$-partitions, Trans. AMS, 349 (1997), pp. 763-788. MR 97f:06006

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 14M15, 05E15, 05E05, 06A07, 14N10


Additional Information

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: https://doi.org/10.1090/S0002-9947-02-02946-X
Received by editor(s): March 7, 2001
Received by editor(s) in revised form: August 6, 2001
Published electronically: 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.
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society