Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T08:12:31.482Z Has data issue: false hasContentIssue false
  • English
  • Français

FRIEZE PATTERNS WITH COEFFICIENTS

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

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; cuntz@math.uni-hannover.de, holm@math.uni-hannover.de
THORSTEN HOLM
Affiliation:
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Welfengarten 1, D-30167Hannover, Germany; cuntz@math.uni-hannover.de, holm@math.uni-hannover.de
PETER JØRGENSEN
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK; peter.jorgensen@ncl.ac.uk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp.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

References

Baur, K., Faber, E., Gratz, S., Serhiyenko, K. and Todorov, G., ‘Friezes satisfying higher $\text{SL}_{k}$-determinants’, Preprint, 2018, arXiv:1810.10562.Google Scholar
Broline, D., Crowe, D. W. and Isaacs, I. M., ‘The geometry of frieze patterns’, Geom. Dedicata 3 (1974), 171–176.CrossRefGoogle Scholar
Conway, J. H. and Coxeter, H. S. M., ‘Triangulated polygons and frieze patterns’, Math. Gaz. 57 (1973), no. 400, 87–94 and no. 401, 175–183.CrossRefGoogle Scholar
Coxeter, H. S. M., ‘Frieze patterns’, Acta Arith. 18 (1971), 297–310.CrossRefGoogle Scholar
Cuntz, M., ‘On wild frieze patterns’, Exp. Math. 26 (2017), 342–348.CrossRefGoogle Scholar
Cuntz, M., ‘A combinatorial model for tame frieze patterns’, Münster J. Math. 12(1) (2019), 49–56.Google Scholar
Cuntz, M. and Holm, T., ‘Frieze patterns over integers and other subsets of the complex numbers’, J. Comb. Algebra 3 (2019), 153–188.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., ‘Cluster algebras I: foundations’, J. Amer. Math. Soc. 15 (2002), 497–529.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., ‘Y-systems and generalized associahedra’, Ann. of Math. (2) 158(3) (2003), 977–1018.CrossRefGoogle Scholar
Holm, T. and Jørgensen, P., ‘A p-angulated generalisation of Conway and Coxeter’s theorem on frieze patterns’, Int. Math. Res. Not. 2020(1) (2020), 71–90.CrossRefGoogle Scholar
Morier-Genoud, S., ‘Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics’, Bull. Lond. Math. Soc. 47(6) (2015), 895–938.CrossRefGoogle Scholar
Morier-Genoud, S. and Ovsienko, V., ‘Farey boat: continued fractions and triangulations, modular group and polygon dissections’, Jahresber. Dtsch. Math.-Ver. 121(2) (2019), 91–136.CrossRefGoogle Scholar
Propp, J., ‘The combinatorics of frieze patterns and Markoff numbers’, Preprint, 2005, arXiv:math/0511633.Google Scholar
Schiffler, R., ‘A cluster expansion formula (A n case)’, Electron. J. Combin. 15(1) (2008), Research paper 64, 9 pages.Google Scholar
Speyer, D. E., ‘Perfect matchings and the octahedron recurrence’, J. Algebraic Combin. 25(3) (2007), 309–348.CrossRefGoogle Scholar
Sylvester, J. J., ‘On the relation between the minor determinants of linearly equivalent quadratic functions’, Philos. Mag., s. IV I (1851), 295–305.CrossRefGoogle Scholar