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      Frieze patterns over integers and other subsets of the complex numbers

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          Abstract

          We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway-Coxeter theory. This model is thus also a model for the set of specializations of cluster algebras of Dynkin type \(A\) in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset \(R\) of the complex numbers. Under certain conditions on \(R\), we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if \(R\) is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over \(R\). Using this, we disprove a conjecture of Fontaine, by showing that for a complex \(d\)-th root of unity \(\zeta_d\) there are only finitely many non-zero frieze patterns for a given height over \(R=\mathbb{Z}[\zeta_d]\) if and only if \(d\in \{1,2,3,4,6\}\).

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          Triangulated polygons and frieze patterns

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            Frieze patterns

            H. Coxeter (1971)
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              Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics

              (2015)
              Frieze patterns of numbers, introduced in the early 70's by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present paper aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.
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                Author and article information

                Journal
                10 November 2017
                Article
                1711.03724
                c6bf2f6d-0004-43e7-b923-068719d03a4d

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                05E15, 05E99, 13F60, 51M20
                25 pages, 10 figures
                math.CO

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