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

  title={Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs},
  author={Lara Bossinger and X. Fang and G. Fourier and Milena Hering and M. Lanini},
  journal={Annals of Combinatorics},
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). 

Figures and Tables from this paper

Toric degenerations of Grassmannians from matching fields
We study the algebraic combinatorics of monomial degenerations of Plucker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky. We provide a necessary condition for aExpand
Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux
An analogue of matching field ideals for Schubert varieties inside the flag variety and a complete characterization of toric ideals among them are described to show that block diagonal matching fields give rise to toric degenerations. Expand
Toric degenerations: a bridge between representation theory, tropical geometry and cluster algebras
In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related. In focus are toric degenerations obtained fromExpand
R T ] 2 8 M ar 2 01 9 Birational sequences and the tropical Grassmannian
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns’ birational sequences and explain how they give rise to points in trop(Gr(k,C)), Speyer-Sturmfels’ tropicalExpand
Lagrangian fillings for Legendrian links of finite type
We prove that there are at least seeds many exact embedded Lagrangian fillings for Legendrian links of type ADE. We also provide seeds many Lagrangian fillings with certain symmetries for type BCFG.Expand
Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian
Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studyingExpand
Birational sequences and the tropical Grassmannian
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$,Expand
Combinatorial mutations and block diagonal polytopes
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties.Expand
Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry
The notion of a Khovanskii basis for $(A, \mathfrak{v})$ is introduced which provides a framework for far extending Gr\"obner theory on polynomial algebras to general finitely generated algeBRas and construct an associated compactification of $Spec(A)$. Expand
Computing toric degenerations of flag varieties
We compute toric degenerations arising from the tropicalization of the full flag varieties \(\mathop{\mathrm{Fl}}\nolimits _{4}\) and \(\mathop{\mathrm{Fl}}\nolimits _{5}\) embedded in a product ofExpand


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