Matching polytopes, toric geometry, and the totally non-negative Grassmannian
TLDR
In this paper, the authors used toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian, denoted (Gr k,n )?0.Abstract:
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.read more
Citations
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KP solitons and total positivity for the Grassmannian
Yuji Kodama,Lauren Williams +1 more
TL;DR: Chakravarty and Chakravarty as discussed by the authors made a connection between the theory of total positivity for the Grassmannian and the structure of regular soliton solutions to the KP equation.
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Totally nonnegative Grassmannian and Grassmann polytopes
TL;DR: The Grassmann polytope was introduced by as discussed by the authors in the context of the current Developments in Mathematics conference in 2014, where Schubert calculus and canonical bases were used to replace linear algebra and convexity in the theory of polytopes.
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Prescriptive Unitarity
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Anatomy of the Amplituhedron
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KP solitons and total positivity for the Grassmannian
Yuji Kodama,Lauren Williams +1 more
TL;DR: In this article, a connection between the theory of total positivity for the Grassmannian and the structure of regular soliton solutions to the dispersive wave equation was made, and it was used to construct all the soliton graphs for (Gr_2n) = 0.
References
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Book
Introduction to Toric Varieties.
TL;DR: In this article, a mini-course is presented to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications, concluding with Stanley's theorem characterizing the number of simplicies in each dimension in a convex simplicial polytope.
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Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.
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Total positivity, Grassmannians, and networks
TL;DR: In this article, the authors discuss the relationship between total positivity and planar directed networks and show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian.