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)



Presenting the cohomology of a Schubert variety

Authors: Victor Reiner, Alexander Woo and Alexander Yong
Journal: Trans. Amer. Math. Soc. 363 (2011), 521-543
MSC (2000): Primary 14M15, 14N15
Published electronically: August 13, 2010
MathSciNet review: 2719692
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton '92]'s definition for permutations. Further refinements are obtained in type $ A$.

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

  • [ALP92] E. Akyildiz, A. Lascoux and P. Pragacz, Cohomology of Schubert subvarieties of $ GL_n/P$. J. Differential Geometry 35 (1992), 511-519. MR 1163446 (93g:14058)
  • [BGG73] I.N. Bernstein, I.M. Gel'fand, S.I. Gel'fand, Schubert cells, and the cohomology of the spaces $ G/P$. Uspehi Mat. Nauk. 28 (1973), no. 3(171), 3-26 [Russian Math. Surveys 28 (1973), no. 3, 1-26]. MR 0429933 (55:2941)
  • [BL00] S. Billey and V. Lakshmibai, Singular loci of Schubert varieties. Progr. Math. 182, Birkhäuser, Boston, 2000. MR 1782635 (2001j:14065)
  • [BB05] A. Björner and F. Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics 231, Springer-Verlag, New York, 2005. MR 2133266 (2006d:05001)
  • [B53] 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. MR 0051508 (14:490e)
  • [C92] J.B. Carrell, Some remarks on regular Weyl group orbits and the cohomology of Schubert varieties, in Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), 33-41. Contemp. Math. 139, Amer. Math. Soc., Providence, RI, 1992. MR 1197828 (93k:14068)
  • [D73] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion. Invent. Math. 21 (1973), 287-301. MR 0342522 (49:7268)
  • [DMR07] M. Develin, J. Martin, V. Reiner, Classification of Ding's Schubert varieties: Finer rook equivalence. Canad. J. Math. 59 (2007), no. 1, 36-62. MR 2289417 (2008b:14080)
  • [D97] K. Ding, Rook placements and cellular decomposition of partition varieties. Discrete Math. 170 (1997), no. 1-3, 107-151. MR 1452940 (98i:05166)
  • [D01] -, Rook placements and classification of partition varieties $ B\backslash M\sb \lambda$. Commun. Contemp. Math. 3 (2001), no. 4, 495-500. MR 1869101 (2003a:05154)
  • [EL96] K. Eriksson and S. Linusson, Combinatorics of Fulton's essential set, Duke Math. J. 85 (1996), 61-76. MR 1412437 (98b:05105)
  • [F92] W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65 (1992), no. 3, 381-420. MR 1154177 (93e:14007)
  • [FP98] W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci. Lecture Notes in Mathematics 1689, Springer-Verlag, Berlin, 1998. MR 1639468 (99m:14092)
  • [GR02] V. Gasharov and V. Reiner, Cohomology of smooth Schubert varieties in partial flag manifolds. J. London Math. Soc. 66 (2002), no. 3, 550-562. MR 1934291 (2003i:14064)
  • [GK97] M. Geck and S. Kim, Bases for the Bruhat-Chevalley order on all finite Coxeter groups. J. Algebra 197 (1997), no. 1, 278-310. MR 1480786 (98k:20066)
  • [H82] H. Hiller, Geometry of Coxeter groups. Research Notes in Mathematics 54, Pitman (Advanced Publishing Program), Boston-London, 1982. MR 649068 (83h:14045)
  • [HLSS07] A. Hultman, S. Linusson, J. Shareshian, and J. Sjöstrand, From Bruhat intervals to intersection lattices and a conjecture of Postnikov. J. Combin. Theory Ser. A 116 (2009), no. 3, 564-580. MR 2500158
  • [H90] J. Humphreys, Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [LS96] A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter. Electron. J. Combin. 3 (1996), no. 2, R27, 35 pp. (electronic). MR 1395667 (98c:05168)
  • [M95] I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford University Press, New York, 1995. MR 1354144 (96h:05207)
  • [M91] -, Notes on Schubert polynomials, Publications du LaCIM. Université du Québec à Montréal, 1991.
  • [M01] L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci. SMF/AMS Texts and Monographs 6, American Mathematical Society, Providence, RI, 2001. MR 1852463 (2002h:05161)
  • [OPY07] S. Oh, A. Postnikov, and H. Yoo Bruhat order, smooth Schubert varieties, and hyperplane arrangements. J. Combin. Theory Ser. A 115 (2008), no. 7, 1156-1166. MR 2450335
  • [R02] N. Reading, Order dimension, strong Bruhat order and lattice properties for posets. Order 19 (2002), no. 1, 73-100. MR 1902662 (2003m:05214)
  • [S01] B.E. Sagan, The symmetric group: Representations, combinatorial algorithms, and symmetric functions, 2nd ed. Graduate Texts in Mathematics 203, Springer-Verlag, New York, 2001. MR 1824028 (2001m:05261)
  • [S99] R.P. Stanley, Enumerative Combinatorics, Volume 2. Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999. MR 1676282 (2000k:05026)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M15, 14N15

Retrieve articles in all journals with MSC (2000): 14M15, 14N15

Additional Information

Victor Reiner
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Alexander Woo
Affiliation: Department of Mathematics, Statistics, and Computer Science, Saint Olaf College, Northfield, Minnesota 55057

Alexander Yong
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Keywords: Schubert calculus, Schubert variety, cohomology presentation, bigrassmannian, essential set
Received by editor(s): November 27, 2008
Received by editor(s) in revised form: June 29, 2009
Published electronically: August 13, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society