Smooth and palindromic Schubert varieties in affine Grassmannians
Sara C. Billey
and Stephen A. Mitchell
DOI: 10.1007/s10801-009-0181-4
Abstract
We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, a chain, one of an infinite family of “spiral” varieties in type A, or a certain 9-dimensional singular variety in type B 3. In particular, except in type A there are only finitely many palindromic affine Schubert varieties in a fixed Lie type. Moreover, in types D and E an affine Schubert variety is smooth if and only if it is palindromic; in all other types there are singular palindromics.
The proofs are for the most part combinatorial. The main tool is a variant of Mozes' numbers game, which we use to analyze the Bruhat order on the coroot lattice. In the proof of the smoothness theorem we also use Chevalley's cup product formula.
Pages: 169–216
Keywords: keywords affine grassmannians; Schubert varieties
Full Text: PDF
References
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2. Billey, S., Mitchell, S.: Affine partitions and the affine Grassmannians (in preparation)
3. Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate Texts in Mathematics, vol.
231. Springer, New York (2005)
4. Bott, R.: An application of the Morse theory to the topology of Lie-groups. Bull. Soc. Math. France 84, 251-281 (1956)
5. Bourbaki, N.: Lie Groups and Lie Algebras. Springer, Berlin (2002)
6. 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, pp. 53-61. Am. Math. Soc., Providence (1994), Part 1
7. Carrell, J., Kuttler, J.: Singular points of t -varieties in g/p and the Peterson map. Invent. Math. 151, 353-379 (2003)
8. Cohen, R., Lupercio, E., Segal, G.: Holomorphic Spheres in Loop Groups and Bott Periodicity. Asian J. Math. 3, 501-518 (1999)
9. Eriksson, K.: Strongly convergent games and Coxeter groups. Ph.D. Thesis, KTH, Stockholm (1993)
10. Humphreys, J.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol.
29. Cambridge University Press, Cambridge (1990)
11. Iwahori, N., Matsumoto, M.: On some Bruhat decomposition and the structure of Hecke rings of p-adic Chevalley groups. Publ. Math. I.H.E.S. 25, 5-48 (1965)
12. Kumar, S.: The nil Hecke ring and singularity of Schubert varieties. Inventiones Math. 123, 471-506 (1996)
13. Kumar, S.: Kac-Moody Groups, their Flag Varieties and Representation Theory. Birkhauser, Boston (2002)
14. Littig, P.: University of Washington Ph.D. Thesis (2005)
15. McCrory, C.: A characterization of homology manifolds. J. London Math. Soc. 16, 146-159 (1977)
16. Mitchell, S.A.: A filtration of the loops on SU(n) by Schubert varieties. Math. Z. 193, 347-362 (1986)
17. Mitchell, S.A.: Parabolic orbits in flag varieties. Preprint (2006).
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