Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties
Abstract
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.
Keywords
Cite
@article{arxiv.1408.0084,
title = {Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties},
author = {Edward Richmond and William Slofstra},
journal= {arXiv preprint arXiv:1408.0084},
year = {2017}
}
Comments
22 pages. Substantial changes for publication; in particular, results for affine type A now appear in arXiv:1702.02236. This version does contain some schematic diagrams which are not in the published version, and which may be helpful