Schubert patches degenerate to subword complexes
Abstract
We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphic to a ball. This gives a new proof of the Andersen-Jantzen-Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points.
Cite
@article{arxiv.0801.4114,
title = {Schubert patches degenerate to subword complexes},
author = {Allen Knutson},
journal= {arXiv preprint arXiv:0801.4114},
year = {2010}
}
Comments
11 pages