Related papers: Frieze patterns as root posets and affine triangul…
We investigate special points on the Grassmannian which correspond to friezes with coefficients in the case of rank two. Using representations of arithmetic matroids we obtain a theorem on subpolygons of specializations of the coordinate…
We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a…
The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last…
We count numbers of tame frieze patterns with entries in a finite commutative local ring. For the ring $\mathbb{Z}/p^r\mathbb{Z}$, $p$ a prime and $r\in\mathbb{N}$ we obtain closed formulae for all heights. These may be interpreted as…
Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. These arrangements are actively studied in connection to the theory of cluster algebras. In the setting of cluster algebras, the notion of a frieze pattern can…
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…
Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains…
Friezes patterns are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their…
Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated…
We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…
Frieze patterns are combinatorial objects that are deeply related to cluster theory. Determinants of frieze patterns arise from triangular regions of the frieze, and they have been considered in previous works by Broline-Crowe-Isaacs, and…
Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an…
The extended affine Weyl group of a root system is the semidirect product of the corresponding Weyl group by its coweight lattice. The stabilizer subgroup of the extended affine Weyl group with respect to the corresponding fundamental…
Following the definition of a root basis of an affine root system, we define a base of the root system of an affine Lie superalgebra to be a linearly independent subset $B$ of its root system such that each root can be written as a linear…
Frieze patterns, as introduced by Coxeter in the 1970's, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in…
Frieze patterns have an interesting combinatorial structure, which has proven very useful in the study of cluster algebras. We introduce $(k,n)$-frieze patterns, a natural generalisation of the classical notion. A generalisation of the…
A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a…
We classify 2-periodic mesh friezes of finite type $A$, $D$ or $E$ with positive real entries. There are families with 0,1, or 2 parameters, depending on type.
We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of…
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…