|
SIGMA 5 (2009), 048, 31 pages arXiv:0809.2081
https://doi.org/10.3842/SIGMA.2009.048
Contribution to the Special Issue on Kac-Moody Algebras and Applications
Singularities of Affine Schubert Varieties
Jochen Kuttler a and Venkatramani Lakshmibai b
a) Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Canada
b) Department of Mathematics, Northeastern University, Boston, USA
Received September 11, 2008, in final form April 03, 2009; Published online April 18, 2009
Abstract
This paper studies the singularities of affine Schubert varieties
in the affine Grassmannian (of type Al(1)). For
two classes of affine Schubert varieties, we determine the
singular loci; and for one class, we also determine explicitly the
tangent spaces at singular points. For a general affine Schubert
variety, we give partial results on the singular locus.
Key words:
Schubert varieties; affine Grassmannian; loop Grassmannian.
pdf (468 kb)
ps (291 kb)
tex (87 kb)
References
- Billey S., Braden T., Lower bounds for Kazhdan-Lusztig polynomials from patterns, Transform. Groups 8 (2003), 321-332, math.RT/0202252.
- Billey S., Mitchell S., Smooth and palindromic Schubert varieties in affine Grassmannians, arXiv:0712.2871.
- Billey S., Postnikov A., Smoothness of Schubert varieties via patterns in root subsystems, Adv. in Appl. Math. 34 (2005), 447-466, math.CO/0205179.
- Billey S., Warrington G., Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 355 (2003), 3915-3945, math.AG/0102168.
- Carrell J., The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, in Algebraic Groups and Their Generalizations: Classical Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Part 1, Amer. Math. Soc., Providence, RI, 1994, 53-61.
- Carrell J., Kuttler J., Smooth points of T-stable varieties in G/B and the Peterson map,
Invent. Math. 151 (2003), 353-379, math.AG/0005025.
- Cortez A., Singularités génériques et quasi-résolutions des variétés de
Schubert pour le groupe linéaire, Adv. Math. 178 (2003), 396-445, math.AG/0106130.
- Evens S., Mircovic I.,
Characteristic cycles for the loop Grassmannian and nilpotent
orbis, Duke Math. J. 97 (1999), 109-126.
- Gasharov V.,
Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties,
Compositio Math. 126 (2001), 47-56.
- Juteau D.,
Modular representations of reductive groups and geometry of affine Grassmannians,
arXiv:0804.2041.
- Kassel C., Lascoux A., Reutenauer C., The singular locus of a Schubert variety, J. Algebra 269 (2003), 74-108.
- Kumar S., The nil Hecke ring and singularity of Schubert varieties, Invent. Math. 123 (1996), 471-506, alg-geom/9503015.
- Kumar S., Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002.
- Kraft H., Procesi C., Minimal singularities in GLn, Invent. Math. 62 (1981), 503-515.
- Lakshmibai V., Sandhya B., Criterion for smoothness of Schubert varieties in
SL(n)/B, Proc. Indian Acad. Sci. Math. Sci. 100 (1990),
45-52.
- Lakshmibai V., Seshadri C.S., Singular locus of a Schubert variety, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 363-366.
- Lakshmibai V., Weyman J., Multiplicities of points on a Schubert variety in a minuscule G/P,
C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 993-996.
- Lusztig G., Green polynomials and singularities of unipotent
classes, Adv. in Math. 42 (1981), 169-178.
- Lusztig G., Canonical bases arising from quantized universal enveloping algebras,
J. Amer. Math. Soc. 3 (1990), 447-498.
- Magyar P.M., Affine Schubert varieties and circular complexes, math.AG/0210151.
- Malkin A., Ostrik V., Vybornov M.,
Minimal degeneration singularities in the affine Grassmannians, Duke Math. J. 126
(2005), 233-249, math.AG/0305095.
- Manivel L., Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices 2001 (2001), no. 16, 849-871, math.AG/0102124.
|
|