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)

 
 

 

Generating varieties for affine Grassmannians


Authors: Peter J. Littig and Stephen A. Mitchell
Journal: Trans. Amer. Math. Soc. 363 (2011), 3717-3731
MSC (2000): Primary 14M15; Secondary 57T99, 55P35
DOI: https://doi.org/10.1090/S0002-9947-2011-05257-8
Published electronically: January 11, 2011
MathSciNet review: 2775825
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study Schubert varieties that generate the affine Grassmannian under the loop group product, and in particular generate the homology ring. There is a canonical such Schubert generating variety in each Lie type. The canonical generating varieties are not smooth, and in fact smooth Schubert generating varieties exist only if the group is not of type $ E_8$, $ F_4$ or $ G_2$.


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

  • 1. Billey, S., and Mitchell, S., Smooth and palindromic Schubert varieties in affine Grassmannians, preprint 55pp: arXiv:0712.2871v1 (2007), to appear in the Journal of Algebraic Combinatorics.
  • 2. Billey, S., and Mitchell, S., Affine partitions and affine Grassmannians, preprint 46pp: arXiv:0712.2871 (2007), to appear in the Electronic Journal of Combinatorics.
  • 3. Bott,R., The space of loops on a Lie group, Mich. Math. J. 5 (1958), 35-61. MR 0102803 (21:1589)
  • 4. Bourbaki, N., Lie Groups and Lie Algebras, Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)
  • 5. Chevalley, C., Sur les décompositions cellulaires des espaces $ G/B$, previously unpublished work appearing in Algebraic Groups and Their Generalizations: Classical Methods, Proc. Symp. Pure Math. 56 (1994), 1-24. MR 1278698 (95e:14041)
  • 6. Evens, S. and Mirković, I., Characteristic cycles for the loop Grassmannian and nilpotent orbits, Duke Math. J. 97 (1999), 109-126. MR 1682280 (2000h:22017)
  • 7. Gutzwiller, L., and Mitchell, S. A., The topology of Birkhoff varieties, Transform. Groups 14 (2009), no. 3, 541-556. MR 2534799
  • 8. Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • 9. Hatcher, A., Algebraic Topology, Cambridge University Press, 2002. MR 1867354 (2002k:55001)
  • 10. Hopkins, M. J., Northwestern University Ph.D. thesis, 1984.
  • 11. Humphreys, J., Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990. MR 1066460 (92h:20002)
  • 12. Iwahori, N., and Matsumoto, H., On some Bruhat decompositions and the structure of Hecke rings of $ p$-adic Chevalley groups, Publ. Math. I.H.E.S. 25 (1965), 5-48. MR 0185016 (32:2486)
  • 13. Juteau, D., Cohomology of the minimal nilpotent orbit, Transformation Groups 13 (2008), 355-387. MR 2426135 (2009g:14058)
  • 14. Kac, V., and Peterson, D., Defining relations of certain infinite-dimensional groups, in Elie Cartan et les mathématiques d'aujourd'hui, Astérisque, hors-série (1985), 165-208. MR 837201 (87j:22027)
  • 15. Kumar, S., Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhauser, Boston 2002. MR 1923198 (2003k:22022)
  • 16. Lam, T., Lapointe, L., Morse, J., and Shimozono, M., Affine insertion and Pieri rules for the affine Grassmannian, preprint: arXiv:math.CO/0609110v2 (2006).
  • 17. Lam, T. Schilling, A., and Shimozono, M., Schubert polynomials for the affine Grassmannian of the symplectic group, Math. Z. 264 (2010), no. 4, 765-811. MR 2593294
  • 18. Magyar, P., Schubert classes of a loop group, preprint: arXiv:0705.3826v1 (2007)
  • 19. Malkin, A., Ostrik, V., and Vybornov, M., The minimal degeneration singularities in the affine Grassmannians, Duke Math. J. 126 (2005), 233-249. MR 2115258 (2005m:14019)
  • 20. Mitchell, S. A., A filtration of the loops on $ \mathrm{SU}(n)$ by Schubert varieties, Math Z. 193 (1986), 347-362. MR 862881 (88d:32043)
  • 21. Mitchell, S. A., The Bott filtration of a loop group, Proc. Barcelona Topology Conference, Lecture Notes in Math. v. 1298, 215-226, Springer-Verlag, 1987. MR 928835 (89d:32070)
  • 22. Mitchell, S. A., Quillen's theorem on buildings and loop groups, L'Enseignement Mathematique 34 (1988), 123-166. MR 960196 (89m:22024)
  • 23. Mitchell, S. A., Parabolic orbits in flag varieties, preprint 2008:1 http://www.math. washington.edu/$ \sim$mitchell/papers/
  • 24. Pressley, A., and Segal, G., Loop Groups, Clarendon Press, Oxford, 1986. MR 900587 (88i:22049)
  • 25. Shafarevich, I., Basic Algebraic Geometry 2 (Schemes and Complex Manifolds), Springer-Verlag, 1994. MR 1328834 (95m:14002)
  • 26. Spanier, E., Algebraic Topology, Springer-Verlag, New York, 1966. MR 666554 (83i:55001)
  • 27. Steinberg, R., Lectures on Chevalley Groups, Yale University Lecture Notes, 1967. MR 0466335 (57:6215)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M15, 57T99, 55P35

Retrieve articles in all journals with MSC (2000): 14M15, 57T99, 55P35


Additional Information

Peter J. Littig
Affiliation: Science and Technology Program, University of Washington, Bothell, Box 358258, Bothell, Washington 98011

Stephen A. Mitchell
Affiliation: Department of Mathematics 354352, University of Washington, Seattle, Washington 98195

DOI: https://doi.org/10.1090/S0002-9947-2011-05257-8
Received by editor(s): November 18, 2008
Received by editor(s) in revised form: November 20, 2009
Published electronically: January 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society