Frieze patterns with coefficients

authored by

Michael Cuntz, Thorsten Holm, Peter Jørgensen

Abstract

Frieze patterns, as introduced by Coxeter in the 1970s, 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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp.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 with 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 (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Newcastle University

Type

Article

Journal

Forum of Mathematics, Sigma

Volume

8

Publication date

26.03.2020

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Computational Mathematics, Analysis, Theoretical Computer Science, Discrete Mathematics and Combinatorics, Geometry and Topology, Algebra and Number Theory, Statistics and Probability, Mathematical Physics

Electronic version(s)

https://doi.org/10.1017/fms.2020.13 (Access: Open)