Regular cell complexes in total positivity

TL;DR

This paper proves a conjecture that certain combinatorial models for Bruhat intervals form regular CW complexes homeomorphic to a ball, using a new criterion linking topology and combinatorics.

Contribution

It establishes that the combinatorial model for Bruhat intervals is a regular CW complex homeomorphic to a ball, confirming a conjecture of Fomin and Shapiro.

Findings

Confirmed the regularity of the CW complex model.

Provided a new criterion for regularity of finite CW complexes.

Connected topology with combinatorial structures in total positivity.

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.