Friezes with coefficients for acyclic cluster algebras

Abstract

We study friezes for cluster algebras of geometric type with coefficients, defined as $\ZZ$-algebra homomorphisms from the cluster algebra to $\ZZ$ with positive integer values on all cluster variables and all frozen variables. When the cluster algebra is acyclic, each acyclic seed defines an acyclic belt of seeds, giving rise to combinatorial objects called {\it frieze patterns with coefficients} which generalise frieze patterns first defined by Coxeter. We give sufficient conditions for friezes to be equivalent to frieze patterns with coefficients. We also show that the choice of coefficients is equivalent to that of a cluster-additive function on the acyclic belt.

Friezes for a finitely generated cluster algebra can be viewed as points on the associated affine variety that have positive integer coordinates in every cluster coordinate chart. For acyclic cluster algebras with trivial coefficients, principal coefficients and BFZ coefficients (named after A. Berenstein, S. Fomin and A. Zelevinsky), we give defining equations for the associated affine varieties in affine space, and describe their frieze points. We also use reduced double Bruhat cells associated to Coxeter elements as geometric models, and describe their frieze points in terms of generalised minors on the associated Kac-Moody groups. When the acyclic cluster algebra is of finite type, we prove that frieze points for principal coefficients are certain {\it integral points} in Lusztig's totally non-negative part of the associated complex simple algebraic group.