相关论文: On the growth of friezes via theta functions

In this article, we study infinite friezes arising from cluster categories of affine type $D$ and determine the growth coefficients for these friezes. We prove that for each affine type $D$, the friezes given by the tubes all have the same…

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…

For a cluster algebra $\mathcal{A}$ over $\mathbb{Q}$ of geometric type, a $\textit{frieze}$ of $\mathcal{A}$ is defined to be a $\mathbb{Q}$-algebra homomorphism from $\mathcal{A}$ to $\mathbb{Q}$ that takes positive integer values on all…

Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call ${\bf Y}\textit{-frieze patterns}$. Using the mutation rules for ${\bf Y}$-variables, we define a large…

We characterize the theta functions for vectors in the imaginary wall in a cluster algebra of acyclic affine type and compute some of their structure constants. One of the structure constant computations can be interpreted as new…

We complete the computation of growth rate of cluster algebras. In particular, we show that growth of all exceptional non-affine mutation-finite cluster algebras is exponential.

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine…

In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup.

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…

We examine the growth behaviour of the entries occurring in $n$-periodic tame friezes of real numbers. Extending \cite{T}, we prove that generalised recursive relations exist between all entries of such friezes. These recursions are…

In \cite{CK2005} and \cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the…

It is known that any infinite frieze comes from a triangulation of an annulus by Baur, Parsons and Tschabold. In this paper we show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way.…

We prove that any entire convex $C^2$-solution to a Hessian type equation with a subquadratic growth at infinity is an affine function.

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…

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…

We extend the recently developed theory of Roehrig and Zwegers on indefinite theta functions to prove certain power series are modular forms. As a consequence, we obtain several power series identities for powers of the generating function…

We prove that the frieze sequences of cluster variables associated with the vertices of an affine quiver satisfy linear recurrence relations. In particular, we obtain a proof of a recent conjecture by Assem-Reutenauer-Smith.

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits…

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…

The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \cite{CK2005} and \cite{SZ} respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras for affine types…