Journal of Algebraic Combinatorics

, Volume 30, Issue 2, pp 173–191

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


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

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


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 Δ G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Δ 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.


Total positivity Grassmannian CW complexes Birkhoff polytope Matching Matroid polytope Cluster algebra

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© Springer Science+Business Media, LLC 2008