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

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dc.creatorPostnikov, Alexander
dc.creatorSpeyer, David
dc.creatorWilliams, Lauren
dc.date2007-06-17
dc.date2008-10-14
dc.date.accessioned2026-07-07T10:09:16Z
dc.date.available2026-07-07T10:09:16Z
dc.descriptionIn 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. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then 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_{kn})_{\geq 0} is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr_{kn})_{\geq 0} is 1.
dc.description26 pages have been reduced to 19 pages, due to: 1. change of coordinates from face to edge variables 2. K. Talaska's theorem in arXiv:0801.4822
dc.identifierhttps://arxiv.org/abs/0706.2501
dc.identifierhttp://arxiv.org/abs/0706.2501
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/171269
dc.subjectAlgebraic Geometry
dc.subjectCombinatorics
dc.subject14P25
dc.titleMatching polytopes, toric geometry, and the non-negative part of the Grassmannian
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