Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs

@article{Bossinger2016ToricDO,
  title={Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs},
  author={Lara Bossinger and Xin Gui Fang and Ghislain Fourier and Milena Hering and Martina Lanini},
  journal={Annals of Combinatorics},
  year={2016},
  volume={22},
  pages={491-512}
}
We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to Gr(2, n). We show that a similar statement does not hold for Gr(3, 6). 

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References

SHOWING 1-10 OF 22 REFERENCES
Birational geometry of cluster algebras
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the LaurentExpand
Khovanskii bases, Newton-Okounkov polytopes and tropical geometry of projective varieties
A Newton-Okounkov body is a convex set associated to a choice of full rank discrete valuation on a commutative algebra. I’ll give a gentle introduction to the construction of Newton-Okounkov bodies,Expand
Grassmannians and Cluster Algebras
This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebraExpand
The tropical Grassmannian
In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedralExpand
On the geometry of binary symmetric models of phylogenetic trees
We investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. We prove that they have Gorenstein terminal singularities and are Fano varieties of index 4 andExpand
The Flag Variety: Geometric and Representation Theoretic Aspects
In this chapter, we discuss how some of the previous results on the Grassmannian apply to the flag variety. We do not provide proofs for all statements included here. For details, the reader mayExpand
Okounkov bodies and toric degenerations
Let $$\varDelta $$ be the Okounkov body of a divisor $$D$$ on a projective variety $$X$$. We describe a geometric criterion for $$\varDelta $$ to be a lattice polytope, and show that in thisExpand
Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
Total positivity, Grassmannians, and networks
The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with theExpand
The Grassmannian Variety: Geometric and Representation-Theoretic Aspects
Preface.- 1. Introduction.- Part I. Algebraic Geometry-A Brief Recollection - 2. Preliminary Material.- 3. Cohomology Theory.- 4. Grobner Bases.- Part II. Grassmannian and Schubert Varieties.- 5. TheExpand
...
1
2
3
...