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

Understand

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.