.svg){kind=logo}
PERMUTATION GROUPS AND FRIEZE PATTERNS OF TYPE A
Keywords:
friezes, cluster algebras, triangulations, permutations, mutationsDOI:
https://doi.org/10.17654/0972555526007Abstract
In this article, we establish a link between permutation groups and frieze patterns of Conway and Coxeter. We define a geometrical mutation of these friezes. We also discuss cluster algebra of type associated with a special permutation called canonical reading.
Received: September 2, 2025
Accepted: November 4, 2025
References
[1] I. Assem and G. Dupont, Friezes and a construction of the Euclidean cluster variables, J. Pure Appl. Algebra 215 (2011), 2322-2340.
[2] I. Assem, G. Dupont, R. Schiffler and D. Smith, Friezes, strings and cluster variables, Glasgow Math. J. 54 (2012), 27-60.
[3] I. Assem and C. Reutenauer, Mutating seeds: types A and Ann. Math. Blaise Pascal 19 (2012), 29-73.
[4] I. Assem, C. Reutenauer and D. Smith, Friezes, Adv. Math. 225 (2010), 3134-3165.
[5] K. Baur, Frieze patterns of integers, Math. Intelligencer 43(2) (2021), 47-54.
[6] K. Baur, E. Faber, S. Gratz, K. Serhiyenko and G. Todorov, Mutation of friezes, Bull. Sci. Math. 142 (2018), 1-48.
[7] P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81(3) (2006), 595-616.
[8] P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters (An case), Trans. Amer. Math. Soc. 358 (2006), 1347-1364.
[9] J. Conway and H. Coxeter, Triangulated polygons and frieze patterns, The Mathematical Gazette 57(400) (1973), 87-94.
[10] J. Conway and H. Coxeter, Triangulated polygons and frieze patterns, The Mathematical Gazette 57(401) (1973), 175-183.
[11] H. Coxeter, Frieze patterns, Acta Arith. 18 (1971), 297-310.
[12] A. G. Elsener and K. Serhiyenko, Mutation of type D friezes, Journal of Combinatorial Theory Series A 176 (2020), 105282.
[13] S. Fomin and N. Reading, Root systems and generalized associahedra Geometric Combinatorics, IAS/Park City Math. Ser., Vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 63-131.
[14] S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces, Part I: Cluster complexes, Acta Mathematica 201(1) (2008), 83-146.
[15] S. Fomin, L. Williams and A. Zelevinsky, Introduction to cluster algebras, chapters 4-5, 2017. arXiv preprint arXiv:1707.07190.
[16] S. Fomin and A. Zelevinsky, Cluster algebras I. Foundations, J. Amer. Math. Soc. 15(2) (2002), 497-529 (in electronic).
[17] S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63-121.
[18] F. Hurtado and M. Noy, Graph of triangulations of a convex polygon and tree of triangulations, Computational Geometry 13(3) 1999, 179-188.
[19] K. E. Magnani, Friezes of type International Electronic Journal of Algebra 17(17) (2015), 66-93.
[20] K. E. Magnani, Friezes of type D, Rev. Un. Mat. Argentina 58(2) (2017), 227-243.
[21] K. E. Magnani, Cluster algebras of type through permutation group Rev. Un. Mat. Argentina 68(1) (2025), 55-68.
[22] G. H. Meisters, Polygons have ears, Am. Math. Mon. 82(6) (1975), 648-651.
[23] S. Morier-Genoud, Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc. 47(6) (2015), 895-938.
[24] M. Pressland, From Frieze Patterns to Cluster Categories, Modern Trends in Algebra and Representation Theory, Volume 486 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2023, pages 109-145.
[25] E. Shalom and C. Lecouvey, Signed permutations and the four color theorem, Expositiones Mathematicae 27(4) (2009), 313-340.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
