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

Postnikov, Alexander ; Speyer, David ; Williams, Lauren

Journal of Algebraic Combinatorics, Tome 30 (2009) no. 2, pp. 173-191.

Résumé

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

Détail

Keywords: keywords total positivity, Grassmannian, CW complexes, Birkhoff polytope, matching, matroid polytope, cluster algebra