Frieze Patterns and Cluster Algebras
Date:
An overview of frieze patterns and their connection to cluster algebras.
A talk for the Bristol Junior Algebra Colloquium.
Abstract
A frieze pattern is a grid of positive integers satisfying a local compatibility condition, known as the \(SL_2\) rule. They were defined in the 1970s by Conway and Coxeter, and we will review some of the results they proved about them. When studying frieze patterns, there is a “Laurent phenomenon” that appears to hold for all friezes.
Another construction with a “Laurent phenomenon” is that of cluster algebras defined by quivers. They were defined in 2002 by Fomin and Zelevinsky, and we will rely on their proof of this phenomenon in the talk. The occurrence of this “Laurent phenomenon” in both constructions is not a coincidence, and we will sketch a proof of a connection between frieze patterns and cluster algebras defined by quivers of type \(A_n\).
(Slides)