Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics. (English) Zbl 1330.05035

Morier-Genoud, Sophie

Bull. Lond. Math. Soc. 47, No. 6, 895-938 (2015).

From the author’s introduction: The present article aims to give an overview of the different approaches and recent results about the notion of friezes. The article is organized in four large independent sections and a short conclusion. In Section 1, we review the original work of Coxeter on frieze patterns [H. S. M. Coxeter, Acta Arith. 18, 297–310 (1971; Zbl 0217.18101)], and discuss some immediate extensions of the results, concerning algebraic properties of friezes and links to projective geometry. In Section 2, we present a generalization of friezes based on representations of quivers and the theory of cluster algebras, introduced in [I. Assem et al., Adv. Math. 225, No. 6, 3134–3165 (2010; Zbl 1275.13017)] and [P. Caldero and F. Chapoton, Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)]. In Section 3, we present the variants of \(\mathrm{SL}_k\)-tilings and \(\mathrm{SL}_k\)-friezes introduced in [F. Bergeron and C. Reutenauer, Ill. J. Math. 54, No. 1, 263–300 (2010; Zbl 1236.13018)]. We review results from [S. Morier-Genoud et al., Forum Math. Sigma 2, Article ID e22, 45 p. (2014; Zbl 1297.39004)], where the space of \(\mathrm{SL}_k\)-friezes is identified with the moduli space of projective polygons and the space of superperiodic difference equations. Section 4 focuses on friezes of positive integers and their relations with different combinatorial models. In particular, we give the Conway-Coxeter correspondence with triangulations of polygons [J. H. Conway and H. S. M. Coxeter, Math. Gaz. 57, 87–94 (1973; Zbl 0285.05028); Math. Gaz. 57, 175–183 (1973; Zbl 0288.05021)] and present some generalizations. The final section lists the different variants of friezes appearing in the literature and some open questions.

Reviewer: Ioan Tomescu (Bucureşti)

MSC:

05B99	Designs and configurations
05E10	Combinatorial aspects of representation theory
13F60	Cluster algebras
14M15	Grassmannians, Schubert varieties, flag manifolds
16G20	Representations of quivers and partially ordered sets
39A70	Difference operators
51M20	Polyhedra and polytopes; regular figures, division of spaces
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Keywords:

Frieze patterns; quiver representations; cluster algebras; projective geometry; superperiodic difference equations; triangulations of polygons; Laurent polynomial; Dynkin quiver; Coxeter number

Citations:

Zbl 0217.18101; Zbl 1275.13017; Zbl 1119.16013; Zbl 1236.13018; Zbl 1297.39004; Zbl 0285.05028; Zbl 0288.05021

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