Friezes

Abstract

The construction of friezes is motivated by the theory of cluster algebras. It gives, for each acyclic quiver, a family of integer sequences, one for each vertex. We conjecture that these sequences satisfy linear recursions if and only if the underlying graph is a Dynkin or an Euclidean (affine) graph. We prove the “only if” part, and show that the “if” part holds true for all non-exceptional Euclidean graphs, leaving a finite number of finite number of cases to be checked. Coming back to cluster algebras, the methods involved allow us to give formulas for the cluster variables in case A m and A ˜ m ; the novelty is that these formulas use 2 by 2 matrices over the semiring of Laurent polynomials generated by the initial variables (which explains simultaneously positivity and the Laurent phenomenon). One tool involved consists of the SL 2 -tilings of the plane, which are particular cases of T-systems of Mathematical Physics.