Matching polytopes, toric geometry, and the non-negative part of the Grassmannian

TL;DR

This paper explores the topology of the non-negative Grassmannian using toric geometry, associating polytopes and toric varieties to its cells, and proving it forms a CW complex with simple cell closures.

Contribution

It introduces a novel connection between polytopes from matchings, matroid polytopes, and toric varieties to analyze the non-negative Grassmannian's topology.

Findings

The cell decomposition of the non-negative Grassmannian is a CW complex.

The Euler characteristic of each cell's closure is 1.

Polytope face lattices are described via matchings of associated graphs.

Abstract

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\geq 0}. This is a cell complex whose cells Delta_G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Delta_G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then use the data of P(G) to define an associated toric variety X_G. We use our technology to prove that the cell decomposition of (Gr_{kn})_{\geq 0} is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr_{kn})_{\geq 0} is 1.