Shelling totally nonnegative flag varieties

TL;DR

This paper proves that the cell decomposition of the totally nonnegative part of any flag variety has a combinatorial structure that is shellable and homeomorphic to a ball, advancing understanding of its topology.

Contribution

The paper establishes that the poset of cells is graded, thin, EL-shellable, and that the associated complex is homeomorphic to a ball, providing new topological insights.

Findings

Q^J is graded, thin, and EL-shellable

The Euler characteristic of each cell closure is 1

The order complex |Q^J| is homeomorphic to a ball

Abstract

In this paper we study the partially ordered set Q^J of cells in Rietsch's cell decomposition of the totally nonnegative part of an arbitrary flag variety P^J_{\geq 0}. Our goal is to understand the geometry of P^J_{\geq 0}: Lusztig has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P^J_{\geq 0} is homeomorphic to a ball. The order complex |Q^J| is a simplicial complex which can be thought of as a combinatorial approximation of P^J_{\geq 0}. Using combinatorial tools such as Bjorner's EL-labellings and Dyer's reflection orders, we prove that Q^J is graded, thin and EL-shellable. As a corollary, we deduce that Q^J is Eulerian and that the Euler characteristic of the closure of each cell is 1. Additionally, our results imply that |Q^J| is homeomorphic to a ball, and moreover, that Q^J is the face poset of some regular CW complex homeomorphic to a ball.