Journal of Algebraic Combinatorics

, Volume 30, Issue 2, pp 173–191

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

Authors

  • Alexander Postnikov
    • Department of MathematicsMassachusetts Institute of Technology
  • David Speyer
    • Department of MathematicsMassachusetts Institute of Technology
    • Department of MathematicsHarvard University
Article

DOI: 10.1007/s10801-008-0160-1

Cite this article as:
Postnikov, A., Speyer, D. & Williams, L. J Algebr Comb (2009) 30: 173. doi:10.1007/s10801-008-0160-1

Abstract

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells ΔG can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell ΔG 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 XG. We use our technology to prove that the cell decomposition of (Grk,n)≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Grk,n)≥0 is 1.

Keywords

Total positivityGrassmannianCW complexesBirkhoff polytopeMatchingMatroid polytopeCluster algebra
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© Springer Science+Business Media, LLC 2008