Combinatorial mutations and block diagonal polytopes

  title={Combinatorial mutations and block diagonal polytopes},
  author={Oliver Clarke and Akihiro Higashitani and Fatemeh Mohammadi},
  journal={arXiv: Combinatorics},
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. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of… Expand

Figures and Tables from this paper

Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs
It is known that the coordinate ring of the Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graphExpand
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
Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties
We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in theExpand
Quadratic Gr\"obner bases of block diagonal matching field ideals and toric degenerations of Grassmannians.
In the present paper, we prove that the toric ideals of certain $s$-block diagonal matching fields have quadratic Grobner bases. Thus, in particular, those are quadratically generated. By using thisExpand
Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Grobner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$Expand


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
Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux
Abstract We study Grobner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux inExpand
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 degeneration of Schubert varieties and Gelfand¿Tsetlin polytopes
This note constructs the flat toric degeneration of the manifold Fln of flags in Cn due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) as an explicit GIT quotient of the GrobnerExpand
Gelfand-Tsetlin polytopes and the integer decomposition property
A natural partial ordering on GT-polytopes is introduced that provides information about integrality and the integer decomposition property, which implies the conjecture for certain shapes. Expand
Minkowski Polynomials and Mutations
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particularExpand
On the f-vectors of Gelfand-Cetlin polytopes
A partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes is obtained, which solves the open problem (2) posed by Gusev, Kritchenko, and Timorin in [GKT]. Expand
Matching fields and lattice points of simplices
Abstract We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matchingExpand
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
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