Details
| Original language | English |
|---|---|
| Article number | e17 |
| Journal | Forum of Mathematics, Sigma |
| Volume | 8 |
| Publication status | Published - 26 Mar 2020 |
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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp.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.
Keywords
- 2010 Mathematics Subject Classification: 13F60 05E15 05E99 51M20
ASJC Scopus subject areas
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Analysis
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Mathematics(all)
- Geometry and Topology
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Statistics and Probability
- Mathematics(all)
- Mathematical Physics
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In: Forum of Mathematics, Sigma, Vol. 8, e17, 26.03.2020.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Frieze patterns with coefficients
AU - Cuntz, Michael
AU - Holm, Thorsten
AU - Jørgensen, Peter
N1 - Funding Information: We are grateful to the anonymous referees for a careful reading of the paper and for numerous useful suggestions. The publication of this article was funded by the Open Access Fund of the Leibniz Universität Hannover.
PY - 2020/3/26
Y1 - 2020/3/26
N2 - 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.
AB - 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.
KW - 2010 Mathematics Subject Classification: 13F60 05E15 05E99 51M20
UR - http://www.scopus.com/inward/record.url?scp=85082316490&partnerID=8YFLogxK
U2 - 10.1017/fms.2020.13
DO - 10.1017/fms.2020.13
M3 - Article
AN - SCOPUS:85082316490
VL - 8
JO - Forum of Mathematics, Sigma
JF - Forum of Mathematics, Sigma
M1 - e17
ER -