Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties

详细信息    查看全文

• 作者：Edward Richmond ; William Slofstra
• 刊名：Mathematische Annalen
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：366
• 期：1-2
• 页码：31-55
• 全文大小：579 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807
• 卷排序：366

文摘

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey–Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson’s theorem that all simply-laced rationally smooth Schubert varieties are smooth. Taken together, our results give a fairly complete geometric description of smooth and rationally smooth Schubert varieties using primarily combinatorial methods.