Matching polytopes, toric geometry, and the totally non-negative Grassmannian
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- Postnikov, A., Speyer, D. & Williams, L. J Algebr Comb (2009) 30: 173. doi:10.1007/s10801-008-0160-1
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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.
KeywordsTotal positivity Grassmannian CW complexes Birkhoff polytope Matching Matroid polytope Cluster algebra
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