Sophie Morier-Genoud
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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Friezes with coefficients are maps assigning numbers to the edges and diagonals of a regular polygon such that all Ptolemy relations for crossing diagonals are satisfied. Among these, the classic Conway-Coxeter friezes are the ones where all values are natural numbers and all edges have value 1. Every subpolygon of a Conway-Coxeter frieze yields a frieze with coefficients over the natural numbers. In this paper we give a complete arithmetic criterion for which friezes with coefficients appear as subpolygons of Conway-Coxeter friezes. This generalizes a result of our earlier paper with Peter Jo…
Coxeter defined the notion of frieze pattern, and Conway and Coxeter proved that triangulations of polygons are in bijection with integral frieze patterns. We show a $p$-angulated generalisation involving non-integral frieze patterns. We also show that polygon dissections give rise to even more general non-integral frieze patterns.
We count the number of Coxeter's friezes over a finite field. Our method uses geometric realizations of the spaces of friezes in a certain completion of the classical moduli space $\mathcal{M}_{0,n}$ allowing repeated points in the configurations. Counting points in the completed moduli space over a finite field is related to the enumeration problem of counting partitions of cyclically ordered set of points into subsets containing no consecutive points. In Appendix we provide an elementary solution for this enumeration problem.
We study a $2 \times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $\mathrm{PSL}(2,\mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
A Conway-Coxeter theorem for decorated frieze patterns
本文引入装饰带状图案的规范化正 Laurent 类，证明其与凸多边形加权三角剖分的典范双射，并给出非递归 Laurent 公式及逆构造方法。
We explain the notion of "$q$-deformed real numbers" introduced in our previous work and overview their main properties. We will also introduce $q$-deformed Conway-Coxeter friezes.
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2\times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
When frieze patterns meet Y-systems: Y-frieze patterns
本文初等介绍Y-饰带图案的概念与性质，并揭示其与Coxeter饰带图案的联系。
In this survey article we explain the intricate links between Conway-Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster tilting object in a cluster category of Dynkin type $A$ in the sense of Caldero and Chapoton.
Conway and Coxeter introduced frieze patterns in 1973 and classified them via triangulated polygons. The determinant of the matrix associated to a frieze table was computed explicitly by Broline, Crowe and Isaacs in 1974, a result generalized 2012 by Baur and Marsh in the context of cluster algebras of type A. Higher angulations of polygons and associated generalized frieze patterns were studied in a joint paper with Holm and Jørgensen. Here we take these results further; we allow arbitrary dissections and introduce polynomially weighted walks around such dissected polygons. The corresponding …
Frieze patterns, as introduced by Coxeter in the 1970's, 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 p…
We consider the variant of Coxeter-Conway frieze patterns called 2-frieze. We prove that there exist infinitely many closed integral 2-friezes (i.e. containing only positive integers) provided the width of the array is bigger than 4. We introduce operations on the integral 2-friezes generating bigger or smaller closed integral 2-friezes.
Arithmetic Y-frieze patterns of width 3 and 4
本文确定了宽度为3和4的全部算术Y型饰带模式，并据此验证了将宽度n的算术Y型饰带模式映射到考克斯特饰带模式的映射p_n在n=3,4时的满射性。
The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last decade has seen a lot of activities on friezes. One reason behind this is the connection to cluster combinatorics.
Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing
本文研究Coxeter带状图案的非交换推广，推广了行列式公式、表述洛朗现象的$T$-路径公式及拼接结果，以非交换版弱带状图案为核心工具。
A frieze is an array of numbers obeying the unimodular rule. Coxeter showed that a frieze with integer entries corresponds to a triangulation. Recently, Holm and Jørgenson introduced friezes of type $Λ_p$ which correspond to $p$-angulations of a polygon. In this paper we explore the connection between these two types of friezes; in particular we show that the friezes of type $Λ_p$ for $p=4$ and $p=6$ contain integral friezes within them. We also consider the relationships with Farey graphs.
Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d-angulation, the determinant is a power of d-1, and the elementary divisors only take values d-1 a…
On continuous 2-frieze patterns
本文研究连续2-条纹图案的定义与性质，将其与射影曲线的模空间及三阶微分算子的Adler-Gelfand-Dikii括号联系起来。
We introduce a new class of friezes which is related to symplectic geometry. On the algebraic and combinatrics sides, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of type ${\rm C}_{2}$ and ${\rm A}_{m}$. On the geometric side, they are related to the moduli space of Lagrangian configurations of points in the 4-dimensional symplectic space introduced in [Conley C.H., Ovsienko V., Math. Ann. 375 (2019), 1105-1145]. Symplectic friezes share similar combinatorial properties to those of Coxeter friezes and SL-friezes.
Given two Coxeter's frieze patterns with the same width and consisting of positive numbers, choose a row and consider the periodic sequence of the differences of the respective entries of the two friezes. We ask for which rows this sequence must change sign at least four times over the period. We prove that this is the case for the first and for the second non-trivial rows, and that this is true, for all rows, for an infinitesimal version of the question. The article also contains expository material on the four vertex theorem and on Coxeter's frieze patterns.
Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspo…
Triangulated polygons and Y-frieze patterns
本文遵循Conway与Coxeter的思路对A型Aₙ的Y-frieze模式分类，解决de Saint Germain关于这类模式源自Conway-Coxeter frieze模式的猜想，还依据Fuss-Catalan数实现了该模式的计数。
Finite and infinite frieze patterns from p-angulations and a generalization of Weyl groupoids
本文提出基于Chebyshev多项式的算法，将p-角化与蛇形图案类型Λ_p联系起来，并扩展了Weyl群oids的定义，揭示了无限蛇形图案与Cartan图的对应关系。
The $q$-deformed cross-ratio: modular invariants and Coxeter friezes
本文引入$q$-形变交比，基于$q$-形变有理数，证明其在模群下不变，并与$q$-形变Coxeter花彩关联，展开得到模不变量序列并计算前两个系数。
Frieze patterns in representation theory
本文探讨了纹饰图案与多边形三角剖分、Grassmannian簇代数及其范畴的联系，并聚焦Grassmannian簇范畴与纹饰图案的最新关联。
Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type A, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter's frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic), and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley-Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley-Me…
Frieze patterns have been introduced by Coxeter in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Canakci and Jorgensen, are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any friez…
We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill's operators. The space of these "superfriezes" is an algebraic supervariety, which is isomorphic to the space of supersymmetric second order difference equations, called Hill's equations.
We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the explicit polynomials arising from the lower bound phenomenon in these quantum cluster algebras.
Frieze patterns over algebraic numbers
本文研究代数数环上的饰带模式，证明代数数环具有有限个单位当且仅当其为负判别式二次域中的序，并进一步刻画了整数环上非零饰带模式何时归结为 Conway-Coxeter 饰带模式。
Friezes and continued fractions
该研究探究Conway和Coxeter提出的数Friezes的基本性质，及其与有理数分解为连分数、Farey序列、模群作用于双曲平面的关系，内容源自2019年俄罗斯杜布纳暑期学校的本科生迷你课程讲义。
Frieze patterns and aperiodic tilings of the plane
研究平面非周期花边图案，基于菱形彭罗斯平铺和戈德雷什 - 兰松 - 比拉尔平铺给出例子，通过用正整数装饰顶点使其满足菱形规则，类似康威和考克斯特定义的图案。
Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Conway-Coxeter frieze pattern. We generalise their result to the corresponding frieze pattern of cluster variables arising from the Fomin-Zelevinsky cluster algebra of type A. We give a representation-theoretic interpretation of this result in terms of certain configurations of indecomposable objects in the root category of type A.
We study the space of 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.
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 questio…
Conway Symbols of Frieze Patterns of Dynkin Types
本文分类了 Dynkin 类型花边图案的 Conway 符号，揭示了其由三角剖分对称性决定，并给出了非单李型及 D 型的具体分类结果。
This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky's cluster algebras of type A. One can derive from the matchings model an enumerative meaning for the Markoff numbers, and prove that the associated Laurent polynomials have positive coefficients as was conjectured (much more…
From frieze patterns to cluster categories
本文以 Conway–Coxeter 带状图案为动机，介绍无圈箭图的丛代数与丛范畴，并用其解释整性与周期性。
Frieze patterns and Farey complexes
本文利用 Farey 复形为任意环上的 frieze 图案建立组合模型，解决了整数模 n 情形的长期公开问题，并推广了直径、计数与提升分类等相关结果。
Geometry of multidimensional Farey summation algorithm and frieze patterns
该研究提出高维减法连分数算法的新几何方法，引入推广Klein多面体的Farey多面体及其帆，用棱柱图分类，得到多维Conway-Coxeter带状图案并证明其满足推广托勒密关系。
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