Matching polytopes, toric geometry, and the totally non-negative Grassmannian

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Gr(k),(n))(>= 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 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(k),(n))(>= 0) is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr(k),(n))(>= 0) is 1.