Frieze patterns in representation theory

Abstract

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 remarkable papers from $1973$, this topic has been nearly forgotten for over thirty years. But since the discovery of connections to cluster algebras and categories of type $A$, interest in friezes has exploded, several generalizations have been studied, and links to geometry and combinatorics have been explored. In this article we will review some of the most striking results connecting the purely combinatorially defined friezes with triangulations of polygons, Grassmannian cluster algebras and (Grassmannian) cluster categories. Then we will focus on Grassmannian cluster categories and some recent results linking them to friezes.