Related papers: Friezes and continuant polynomials with parameters
We introduce a class of commutative superalgebras generalizing cluster algebras. A cluster superalgebra is defined by a hypergraph called an "extended quiver", and transformations called mutations. We prove the super analog of the "Laurent…
We define the notion of infinite friezes of positive integers as a variation of Conway-Coxeter frieze patterns and study their properties. We introduce useful gluing and cutting operations on infinite friezes. It turns out that…
We consider frieze sequences corresponding to sequences of cluster mutations for affine D and E type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient…
Let $Q$ be an euclidean quiver. Using friezes in the sense of Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical) cluster character associated to any object in the cluster category of $Q$. In particular, this…
By viewing $\tilde{A}$ and $\tilde{D}$ type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze pattern (or bipartite belt) or (ii) the periodic quantities previously found for the…
In this article, we establish a link between the values of a frieze of type D and some values of a particular frieze of type A. This link allows us to compute, independently of each other, all the cluster variables in the cluster algebra…
A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with…
It is an important aspect of cluster theory that cluster categories are "categorifications" of cluster algebras. This is expressed formally by the (original) Caldero-Chapoton map X which sends certain objects of cluster categories to…
Frieze patterns have been introduced by Coxeter in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values…
This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the…
We analyse the growth coefficients of infinite frieze patterns arising from cluster algebras using cluster modular groups and cluster categories. For a fixed cluster category of affine type, we prove that the collection of infinite frieze…
The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps "reachable" indecomposable objects to the corresponding cluster…
Motivated by a construction in the theory of cluster algebras (Fomin and Zelevinsky), one associates to each acyclic directed graph a family of sequences of natural integers, one for each vertex; this construction is called a {\em frieze};…
Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper…
Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a…
We consider the variant of Coxeter-Conway frieze patterns called 2-frieze. We prove that there exist infinitely many closed integral 2-friezes (i.e. containing only positive integers) provided the width of the array is bigger than 4. We…
Coxeter defined the notion of frieze pattern, and Conway and Coxeter proved that triangulations of polygons are in bijection with integral frieze patterns. We show a $p$-angulated generalisation involving non-integral frieze patterns. We…
Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as…
In this article, we construct SL$_k$-friezes using Pl\"ucker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Pl\"ucker embedding. When this cluster…
In this paper, we provide a combinatorial interpretation for Laurent polynomials obtained by iteratively mutating a certain periodic quiver that has been framed with frozen vertices. This yields a family of cluster variables with principal…