Double Schubert polynomials and degeneracy loci for the classical groups

Andrew Kresch[1]; Harry Tamvakis[2]

  • [1] University of Pennsylvania, Department of Mathematics, Philadelphia PA 19104-6395 (USA)
  • [2] Brandeis University, Department of Mathematics, Waltham MA 02454-9110 (USA)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 6, page 1681-1727
  • ISSN: 0373-0956

Abstract

top
We propose a theory of double Schubert polynomials P w ( X , Y ) for the Lie types B , C , D which naturally extends the family of Lascoux and Schützenberger in type A . These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When w is a maximal Grassmannian element of the Weyl group, P w ( X , Y ) can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type A formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows there are no Chern class formulas for degeneracy loci of “isotropic morphisms” of bundles.

How to cite

top

Kresch, Andrew, and Tamvakis, Harry. "Double Schubert polynomials and degeneracy loci for the classical groups." Annales de l’institut Fourier 52.6 (2002): 1681-1727. <http://eudml.org/doc/116024>.

@article{Kresch2002,
abstract = {We propose a theory of double Schubert polynomials $P_w(X,Y)$ for the Lie types $B$, $C$, $D$ which naturally extends the family of Lascoux and Schützenberger in type $A$. These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When $w$ is a maximal Grassmannian element of the Weyl group, $P_w(X,Y)$ can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type $A$ formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows there are no Chern class formulas for degeneracy loci of “isotropic morphisms” of bundles.},
affiliation = {University of Pennsylvania, Department of Mathematics, Philadelphia PA 19104-6395 (USA); Brandeis University, Department of Mathematics, Waltham MA 02454-9110 (USA)},
author = {Kresch, Andrew, Tamvakis, Harry},
journal = {Annales de l’institut Fourier},
keywords = {degeneracy loci; Schubert polynomials; determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes},
language = {eng},
number = {6},
pages = {1681-1727},
publisher = {Association des Annales de l'Institut Fourier},
title = {Double Schubert polynomials and degeneracy loci for the classical groups},
url = {http://eudml.org/doc/116024},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Kresch, Andrew
AU - Tamvakis, Harry
TI - Double Schubert polynomials and degeneracy loci for the classical groups
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 6
SP - 1681
EP - 1727
AB - We propose a theory of double Schubert polynomials $P_w(X,Y)$ for the Lie types $B$, $C$, $D$ which naturally extends the family of Lascoux and Schützenberger in type $A$. These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When $w$ is a maximal Grassmannian element of the Weyl group, $P_w(X,Y)$ can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type $A$ formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows there are no Chern class formulas for degeneracy loci of “isotropic morphisms” of bundles.
LA - eng
KW - degeneracy loci; Schubert polynomials; determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes
UR - http://eudml.org/doc/116024
ER -

References

top
  1. E. Akyildiz, J. Carrell, An algebraic formula for the Gysin homomorphism from G / B to G / P , Illinois J. Math 31 (1987), 312-320 Zbl0629.57030MR882116
  2. N. Bergeron, F. Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J 95 (1998), 373-423 Zbl0939.05084MR1652021
  3. I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Schubert cells and cohomology of the spaces G / P , Russian Math. Surveys 28 (1973), 1-26 Zbl0289.57024MR429933
  4. A. Bertram, Quantum Schubert calculus, Adv. Math 128 (1997), 289-305 Zbl0945.14031MR1454400
  5. S. Billey, Kostant polynomials and the cohomology ring for G / B , Duke Math. J 96 (1999), 205-224 Zbl0980.22018MR1663931
  6. S. Billey, M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc 8 (1995), 443-482 Zbl0832.05098MR1290232
  7. A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math 57 (1953), 115-207 Zbl0052.40001MR51508
  8. I. Ciocan, - Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc 351 (1999), 2695-2729 Zbl0920.14027MR1487610
  9. M. Demazure, Invariants symétriques des groupes de Weyl et torsion, Invent. Math 21 (1973), 287-301 Zbl0269.22010MR342522
  10. M. Demazure, Désingularization des variétés de Schubert généralisées, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 7 (1974), 53-88 Zbl0312.14009MR354697
  11. S. Fomin, A. N. Kirillov, Combinatorial B n -analogs of Schubert polynomials, Trans. Amer. Math. Soc 348 (1996), 3591-3620 Zbl0871.05060MR1340174
  12. W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J 65 (1992), 381-420 Zbl0788.14044MR1154177
  13. W. Fulton, Schubert varieties in flag bundles for the classical groups, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) (1996), 241-262 Zbl0862.14032
  14. W. Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom 43 (1996), 276-290 Zbl0911.14001MR1424427
  15. W. Fulton, Intersection Theory, 2 (1998), Springer-Verlag, Berlin Zbl0885.14002MR1644323
  16. W. Fulton, P. Pragacz, Schubert varieties and degeneracy loci, 1689 (1998), Springer-Verlag, Berlin Zbl0913.14016MR1639468
  17. W. Graham, The class of the diagonal in flag bundles, J. Differential Geom 45 (1997), 471-487 Zbl0935.14015MR1472885
  18. A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki 13 (1960/61) Zbl0236.14003
  19. J. Harris, L. W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), 71-84 Zbl0534.55010MR721453
  20. T. Józefiak, A. Lascoux, P. Pragacz, Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Math. USSR Izvestija 18 (1982), 575-586 Zbl0489.14020MR623355
  21. G. Kempf, D. Laksov, The determinantal formula of Schubert calculus, Acta Math 132 (1974), 153-162 Zbl0295.14023MR338006
  22. A. Kresch, H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian Zbl1051.53070MR1993764
  23. A. Kresch, H. Tamvakis, Quantum cohomology of orthogonal Grassmannians, (2001) Zbl1077.14083MR2027200
  24. A. Lascoux, Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris, Sér. I Math 295 (1982), 393-398 Zbl0495.14032MR684734
  25. A. Lascoux, P. Pragacz, Operator calculus for Q ˜ -polynomials and Schubert polynomials, Adv. Math 140 (1998), 1-43 Zbl0951.14035MR1656481
  26. A. Lascoux, P. Pragacz, Orthogonal divided differences and Schubert polynomials, P ˜ -functions, and vertex operators, Michigan Math. J 48 (2000), 417-441 Zbl1003.05106MR1786499
  27. A. Lascoux, P. Pragacz, Schur Q -functions and degeneracy locus formulas for morphisms with symmetries, (2000), 239-263, Birkhäuser, Boston Zbl0969.14033
  28. A. Lascoux, M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris, Sér. I Math 294 (1982), 447-450 Zbl0495.14031MR660739
  29. I. G. Macdonald, Notes on Schubert polynomials, 6 (1991), Publ. LACIM, Univ. de Québec à Montréal, Montréal Zbl0784.05061MR1161461
  30. I. G. Macdonald, Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991) 166 (1991), 73-99, Cambridge Univ. Press, Cambridge Zbl0784.05061
  31. I. G. Macdonald, Symmetric Functions and Hall Polynomials, (1995), Clarendon Press, Oxford Zbl0824.05059MR1354144
  32. P. Pragacz, Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Topics in Algebra, Part 2 (Warsaw, 1988) 26 (1990), 189-199, Banach Center Publ, Part 2, PWN, Warsaw Zbl0743.14009
  33. P. Pragacz, Algebro-geometric applications of Schur S - and Q -polynomials, Séminaire d'Algèbre Dubreil-Malliavin 1989-1990 1478 (1991), 130-191, Springer-Verlag, Berlin Zbl0783.14031
  34. P. Pragacz, J. Ratajski, A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians, J. reine angew. Math 476 (1996), 143-189 Zbl0847.14029MR1401699
  35. P. Pragacz, J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; Q ˜ -polynomial approach, Compositio Math 107 (1997), 11-87 Zbl0916.14026MR1457343
  36. I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math 139 (1911), 155-250 Zbl42.0154.02
  37. H. Tamvakis, Arakelov theory of the Lagrangian Grassmannian, J. reine angew. Math 516 (1999), 207-223 Zbl0934.14018MR1724621
  38. L. W. Tu, Degeneracy loci, Proc. conf. algebraic geom. (Berlin, 1985) 92 (1986), 296-305, Teubner, Leipzig Zbl0626.14019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.