Pattern characterization of rationally smooth affine Schubert varieties of type A
- 作者：Sara Billey billey@uw.edu ; Andrew Crites ; acrites@uw.edu
- 关键词：Pattern avoidance ; Affine permutations ; Schubert varieties
- 刊名：Journal of Algebra
- 出版年：2012
- 期刊代码：69_00218693
- 类别：mt
- 出版时间：1 July, 2012
- 卷：361
- 期：Complete
- 页码：107-133
- 文件大小：656 K
摘要
Schubert varieties in finite dimensional flag manifolds are a well-studied family of projective varieties indexed by elements of the corresponding Weyl group W. In particular, there are many tests for smoothness and rational smoothness of these varieties. One key result due to Lakshmibai and Sandhya is that in type A the smooth Schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412. Recently, there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with G being a Kac-Moody group and W being an affine Weyl group or parabolic quotient. In this paper we study the case when W is the affine Weyl group of type A or the affine permutations. We develop the notion of pattern avoidance for affine permutations. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations.