Related papers: Newton polytopes of rank 3 cluster variables
The Helton-Nie Conjecture (HNC) is the proposition that every convex semialgebraic set is a spectrahedral shadow. Here we prove that HNC is equivalent to another propo- sition related to quadratically constrained quadratic programming.…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
Let $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ be a weighted projective line. We define the quantum cluster algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ and realize its specialized version as the subquotient of the…
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group…
Let $G=(V, E)$ be an undirected graph. The spanning trees polytope $P(G)$ is the convex hull of the characteristic vectors of all spanning trees of $G$. In this paper, we describe all facets of $P(G)$ as a consequence of the facets of the…
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…
For any given finite group, Schulte and Williams (2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the…
We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the…
Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $\Sigma$ with certain…
We provide a concrete realization of the cluster algebras associated with Q-systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply-laced groups, this provides a cluster-algebraic…
A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities,…
We prove that the weight polytope of the Hurwitz form of a polarized smooth toric variety coincides with the convex hull of the characteristic vectors introduced in the previous work of Ogusu and the author with respect to all regular…
There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${{\boldsymbol{\mathsf U}}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations…
We studies the Newton polygon for the L-function of toric exponential sums attached to a family of two variable generalized hyperkloosterman sum,$f_{t}(x,y)=x^{n}+y+\frac{t}{xy}$ with $t$ the parameter. The explicit Newton polygon is…
We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the cluster-tilting objects form one equivalence class under mutation.
We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give…
The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a…
We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a…
Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 quiver and geometric properties of its corresponding…