FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS

Description

2-FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO, AND SERGE TABACHNIKOV Abstract. 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. Contents 1. Introduction 2 1.1. 2-friezes 2 1.2. Geometric version: moduli spaces of n-gons 3 1.3. Analytic version: the space of difference equations 4 1.4. The pentagram map and cluster structure 5 2. Definitions and main results 5 2.1. Algebraic and numerical friezes 5 2.2. Closed frieze patterns 6 2.3. Closed 2-friezes, difference equations and n-gons 7 2.4. Cluster structure 7 2.5. Arithmetic 2-friezes 8 3. Algebraic 2-friezes 9 3.1. The pattern rule 9 3.2. The determinant formula 10 3.3. Recurrence relations on the diagonals 10 3.4. Relation to SL3-tilings 11 4. Numerical friezes 12 4.1. Entries of a numerical frieze 12 4.2. Proof of Proposition 2.3 12 4.3.

• recurrence


• dimensional vector


• frieze patterns


• n? 8)-dimensional algebraic


• recurrence relations


• closed


• friezes


• f0n ?


• width n?


• all known

Sujets

• Récurrence
• Recurrence relation
• Closed
• Frieze

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15
• Langue: English

Extrait

2-FRIEZE PATTERNS AND THE CLUSTER STRUCTURE THE SPACE OF POLYGONS

OF

SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO, AND SERGE TABACHNIKOV

Abstract.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 ofn-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves withn show that the space of 2-frieze patternsmarked points. We is a cluster manifold and study its algebraic and arithmetic properties.

Contents

1. Introduction 1.1. 2-friezes 1.2. Geometric version: moduli spaces ofn-gons 1.3. Analytic version: the space of difference equations 1.4. The pentagram map and cluster structure 2. Definitions and main results 2.1. Algebraic and numerical friezes 2.2. Closed frieze patterns 2.3. Closed 2-friezes, difference equations andn-gons 2.4. Cluster structure 2.5. Arithmetic 2-friezes 3. Algebraic 2-friezes 3.1. The pattern rule 3.2. The determinant formula 3.3. Recurrence relations on the diagonals 3.4. Relation to SL 3 -tilings 4. Numerical friezes 4.1. Entries of a numerical frieze 4.2. Proof of Proposition 2.3 4.3. Proof of Theorem 1 4.4. Difference equations, and polygons in space and in the projective plane 4.5. Convex polygons in space and in the projective plane 4.6. The spaceC 3m and the Fock-Goncharov variety 5. Closed 2-friezes as cluster varieties 5.1. Cluster algebras 5.2. The algebra of regular functions onF n 5.3. Zig-zag coordinates 5.4. The cluster manifold of closed 2-friezes 5.5. The symplectic structure 6. Arithmetic 2-friezes

Key words and phrases.Pentagram map, Cluster algebra, Frieze pattern, Moduli space. 1

2 2 3 4 5 5 5 6 7 7 8 9 9 10 10 11 12 12 12 13 14 15 15 16 16 17 18 20 21 23

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SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO, AND SERGE TABACHNIKOV

6.1. Arithmetic 2-friezes forn= 4,5 6.2. Arithmetic 2-friezes forn= 6 6.3. One-point stabilization procedure 6.4. Connected sum 6.5. Examples of infinite arithmetic 2-frieze patterns Appendix: Frieze patterns of Coxeter-Conway, difference equations, polygons, and the moduli spaceM 0,n References

1.rontctduoiIn

23 24 25 27 29

31 34

The spaceC n ofn-gons in the projective plane (overCor overR) modulo projective equiv- alence is a close relative of the moduli spaceM 0,n of genus zero curves withnmarked points. The spaceC n was considered in [25] and in [20] as the space on which thepentagram mapacts. The main idea of this paper is to identify the spaceC n with the spaceF n of combinatorial objects that we call 2-friezesappeared in [22] as generalization of the Cox-. These objects first eter friezes [6]. We show thatC n is isomorphic toF n , providedn Thisis not a multiple of 3. isomorphism leads to remarkable coordinate systems onC n and equipsC n with the structure of cluster manifold. The relation between 2-friezes and cluster algebras is not surprising, since 2-friezes can be viewed as a particular case of famous recurrence relations known as the dis- crete Hirota equation, or the octahedron recurrence. The particular case of 2-friezes is a very interesting subject; in this paper we make first steps in study of algebraic and combinatorial structures of the space of 2-friezes. The pentagram mapT:C n → C n , see [24, 25] and also [20, 16], is a beautiful dynamical system which is a time and space discretization of the Boussinesq equation. Integrability of TonC n is still an open problem (integrability was proved [20] for a larger space of twisted n-gons). Thestructure of the space of closed polygons was our desire to better understand the main motivation.

1.1. 2-friezes.We call a2-frieze patterna grid of numbers, or polynomials, rational functions, etc., (v i,j ) (i,j)∈Z2 and (v i+ 2 1 ,j+ 1 2 ) (i,j)∈Z2 organized as follows M M M v q i− 2 1 ,j+ 2 1 M q q q q M M M M q M M q M M q q q q q qq  M q M M q q M 1 . . . M M M M M M M M M M q v M q i M − 1 2 ,j− M 1 2 q M M M q q v i,j M q M v i+ 2 1 ,j+ K 2 sss q s q q q q q q q M q q q M M M M M M M M q K K s s s q s q q  q  MMM  sss  K  K s v i− 1 2 ,j− 3 2 v i,j−1 v i+ 2 1 ,j− 2 1 v i+1,j ∙ ∙ ∙ s M q M q K K s K K K s K q M q q M q q M q q q M q q M q q M M q M M q M M q M M M M q q q M q q M q q q M q q M s s K K K K v i,j−2 v i+ 2 1 ,j− 3 2 v i+1,j−1 v i+ 2 3 ,j− 1 2 v i+2,j q K s s M M M M M q M q q q q M M M q q K s K s s s q M q M q M q M M M q q q M q q q M q M M q q M M M M M M M M s s s s K K q v i+1,j−2 v i+ 2 3 ,j− 2 3 v i+2,j−1 K M q q q M q M M M q M M q M M M q q q q M q q M M M q q q q M M q v i+2,j−2 M

2-FRIEZE PATTERNS AND THE SPACE OF POLYGONS

3

such that every entry is equal to the determinant of the 2×2-matrix formed by its four neigh- bours:

B F ? ? ?    ? ?    @ ~ ~ ~ ~ ~ ~ ~ @      ? ?  ? ? ? ? ? ~ ~ @ ? @ A E D H ? ? ? ?   ? ?   @ @ ~ ? ? ?    @ ~ @ @ ~  ?   ? ~ @ @   C  ? G @ @

E=AD−BC, D=EH−F G, . . .

Generically, two consecutive rows in a 2-frieze pattern determine the whole 2-frieze pattern. The notion of 2-frieze pattern is an analog of the classical notion of Coxeter-Conway frieze pattern [6, 5]. Similarly to the classical frieze patterns, 2-frieze patterns constitute a particular case of the 3-dimensional octahedron recurrence:

T i+1,j,k T i−1,j,k =T i,j+1,k T i,j−1,k −T i,j,k+1 T i,j,k−1 , which may be called the Dodgson condensation formula (1866) and which is also known in the mathematical physics literature as the discrete Hirota equation (1981). More precisely, assume T −1,j,k =T 2,j,k = 1 andT i,j,k = 0 fori≤ −2 andi≥3. ThenT 0,j,k andT 1,j,k form a 2-frieze. More general recurrences called theT-systems and their relation to cluster algebras were studied recently, see [17, 9, 18] and references therein. In particular, periodicity and positivity results, typical for cluster algebras, were obtained. The above 2-frieze rule was mentioned in [22] as a variation on the Coxeter-Conway frieze pattern. What we call a 2-frieze pattern also appeared in [2] in a form of duality on SL 3 -tilings. To the best of our knowledge, 2-frieze patterns have not been studied in detail before. We are particularly interested in 2-frieze patterns bounded from above and from below by two rows of 1’s: ∙ ∙ ∙1 1 1 1 1∙ ∙ ∙ 1 3 v 2 2 ∙ ∙ ∙ ∙ ∙ ∙v 0,0 v 1 2 , 2 v 1,1 v 2 3 , 2 , . . . . . ∙ ∙ ∙1 1 1 1 1∙ ∙ ∙ that we callclosed We call the2-frieze patterns.widthof a closed pattern the number of the rows between the two rows of 1’s. We introduce the following notation:

F n ={closed 2-friezes of widthn−4} for the space of all closed (complex or real) 2-frieze patterns. Here and below the term “space” is used to identify a set of objects that we wish to endow with a geometric structure of (algebraic, smooth or analytic) variety. We denote byF n 0 ⊂ F n subspace of closed friezes of widththe n−4 such that all their entries arereal positive. Along with the octahedron recurrence, the space of all 2-frieze patterns is closely related to the theory of cluster algebras and cluster manifolds [12]. In this paper, we explore this relation.

1.2. spaces ofGeometric version: modulin-gons.Ann-gonin the projective plane is given by a cyclically orderedn-tuple of points{v 1 , . . . , v n }inP 2 such that no three consecutive points belong to the same projective line. In particular,v i 6=v i+1 , andv i 6=v i+2  one. However, may havev i =v j , if|i−j| ≥3. We understand then-tuple{v 1 , . . . , v n }as an infinite cyclic sequence, that is, we assumev i+n =v i , for alli= 1, . . . , n.