Related papers: Cluster expansion formulas in type A
Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient…
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…
We introduce a handy construction of cluster algebras of type $\mathbb{A}_{\infty}$, we give a complete classification of the cluster algebras arising from the infinity-gon, and finally we construct the category of the diagonals of the…
The goals of this expository article are on one hand to describe how to construct ($m$-) cluster categories from triangulations (resp. from $m+2$-angulations) of polygons. On the other hand, we explain how to use translation quivers and…
We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several…
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a…
The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and…
Let $Q$ be any invertible valued quiver without oriented cycles. We study connections between the category of valued representations of $Q$ and expansions of cluster variables in terms of the initial cluster in quantum cluster algebras. We…
Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to…
We show that for $2\le d\le 4$, every finite geometric simplicial complex $\Delta$ in $\mathbb{R}^d$ with vertices on the moment curve can be extended to a triangulation $T$ of the cyclic polytope $C$ where $\Delta, T$ and $C$ all have the…
This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds…
Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider the torus of genus n with two interior…
We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that…
We give an explicit combinatorial formula for the Laurent expansion of any arc or closed curve on an unpunctured triangulated orbifold. We do this by extending the snake graph construction of Musiker, Schiffler, and Williams to unpunctured…
This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a…
We present an effective method for recovering the topology of a bordered oriented surface with marked points from its cluster algebra. The information is extracted from the maximal triangulations of the surface, those that have exchange…
The cluster expansion formalism used in materials science is reconstructed on an axiomatic basis with the aims of clarifying underlying concepts and improving computational procedures, and without using conventional cluster functions.…
We provide a cluster-algebraic approach to the computation of the recently introduced generalised biadjoint scalar amplitudes related to Grassmannians ${\rm Gr}(k,n)$. A finite cluster algebra provides a natural triangulation for the…
The cluster multiplication formulas for a generalized quantum cluster algebra of Kronecker type are explicitly given. Furthermore, a positive bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-basis of this algebra is constructed.
A short exposition with complete proofs of the theory of cluster expansions for an abstract polymer system is presented.