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FRIEZE PATTERNS WITH COEFFICIENTS

Part of: Algebraic combinatorics Real and complex geometry Arithmetic rings and other special rings

Published online by Cambridge University Press:  26 March 2020

MICHAEL CUNTZ ,
THORSTEN HOLM  and
PETER JØRGENSEN
MICHAEL CUNTZ
Affiliation:
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Welfengarten 1, D-30167Hannover, Germany; [email protected], [email protected]
THORSTEN HOLM
Affiliation:
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Welfengarten 1, D-30167Hannover, Germany; [email protected], [email protected]
PETER JØRGENSEN
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK; [email protected]

Abstract

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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp.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

MSC classification

Primary: 13F60: Cluster algebras
Secondary: 05E15: Combinatorial aspects of groups and algebras 05E99: None of the above, but in this section 51M20: Polyhedra and polytopes; regular figures, division of spaces
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e17
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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