Regular cell complexes in total positivity
Combinatorics
2013-07-08 v5 Geometric Topology
Abstract
This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell intersected with a larger closed cell, all within the totally nonnegative part of the unipotent radical of an algebraic group. A parametrization due to Lusztig turns out to have all the requisite features to provide the attaching maps. A key ingredient is a new, readily verifiable criterion for which finite CW complexes are regular involving an interplay of topology with combinatorics.
Cite
@article{arxiv.0711.1348,
title = {Regular cell complexes in total positivity},
author = {Patricia Hersh},
journal= {arXiv preprint arXiv:0711.1348},
year = {2013}
}
Comments
accepted to Inventiones Mathematicae; 60 pages; substantially revised from earlier versions