Frieze patterns with coefficients

TL;DR

This paper systematically studies generalized frieze patterns with coefficients related to cluster algebras, extending classic results, and characterizes their construction and finiteness properties.

Contribution

It introduces a comprehensive framework for frieze patterns with coefficients, generalizing classical results and providing a complete characterization for certain cases.

Findings

Frieze patterns with coefficients can be derived from classic frieze patterns by cutting subpolygons.

Complete characterization of frieze patterns with coefficients for triangles.

Finiteness of such patterns with entries in a discrete subset of complex numbers.

Abstract

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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated to classic Conway-Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (non-zero) frieze patterns with coefficients with entries in a discrete subset of the complex numbers.