Frieze patterns with coefficients

verfasst von

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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp.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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

Newcastle University

Typ

Artikel

Journal

Forum of Mathematics, Sigma

Band

8

Publikationsdatum

26.03.2020

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Computational Mathematics, Analysis, Theoretische Informatik, Diskrete Mathematik und Kombinatorik, Geometrie und Topologie, Algebra und Zahlentheorie, Statistik und Wahrscheinlichkeit, Mathematische Physik

Elektronische Version(en)

https://doi.org/10.1017/fms.2020.13 (Zugang: Offen)