2-frieze patterns and the cluster structure of the space of polygons

Morier-Genoud, Sophie; Ovsienko, Valentin; Tabachnikov, Serge

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 937-987.

Nous étudions une variante des frises de Coxeter-Conway appelée 2-frises. La réalisation géométrique de l’espace des 2-frises est l’espace des modules de polygones, dans le plan projectif ou dans l’espace vectoriel de dimension 3, qui est un analogue de l’espace des modules des courbes de genre 0 avec $n$ points marqués. Nous montrons que l’espace des 2-frises admet une structure de variété amassée et nous en étudions les propriétés algébriques et arithmétiques.

We study 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.

Classification : 13F60, 14N05, 51M99
Keywords: Frieze patterns, Coxeter-Conway friezes, moduli space, cluster algebra, pentagram map.
Mot clés : Frises, frises de Coxeter-Conway, espace de modules, algebre amassée, application pentagramme.

[1] Aguirre, L.; Felder, G.; Veselov, A. Gaudin subalgebras and stable rational curves (arXiv:1004.3253)

[2] Bergeron, F.; Reutenauer, C. ${\mathrm{SL}}_k$-Tiling of the Plane, Illinois J. Math., Volume 54 (2010), pp. 263-300 | MR

[3] Caldero, P.; Chapoton, F. Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Volume 81 (2006), pp. 595-616 | DOI | MR

[4] Chapoton, F. (Unpublished notes)

[5] Conway, J. H.; Coxeter, H. S. M. Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973), p. 87-94 and 175–183 | DOI | MR | Zbl

[6] Coxeter, H. S. M. Frieze patterns, Acta Arith., Volume 18 (1971), pp. 297-310 | MR | Zbl

[7] Di Francesco, P. The solution of the $A_r$ T-system for arbitrary boundary, Electron. J. Combin., Volume 17 (2010) no. 1 (Research Paper 89, 43 pp) | MR

[8] Di Francesco, P.; Kedem, R. Positivity of the $T$-system cluster algebra, Electron. J. Combin., Volume 16 (2009) | MR

[9] Di Francesco, P.; Kedem, R. $Q$-systems as cluster algebras. II. Cartan matrix of finite type and the polynomial property, Lett. Math. Phys., Volume 89 (2009), pp. 183-216 | DOI | MR

[10] Fock, V.; Goncharov, A. Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Etudes Sci., Volume 103 (2006), pp. 1-211 | EuDML | Numdam | MR | Zbl

[11] Fock, V.; Goncharov, A. Moduli spaces of convex projective structures on surfaces, Adv. Math., Volume 208 (2007), pp. 249-273 | DOI | MR | Zbl

[12] Fomin, S.; Zelevinsky, A. Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002), pp. 497-529 | DOI | MR | Zbl

[13] Fomin, S.; Zelevinsky, A. The Laurent phenomenon, Adv. in Appl. Math., Volume 28 (2002), pp. 119-144 | DOI | MR | Zbl

[14] Fomin, S.; Zelevinsky, A. Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007), pp. 112-164 | DOI | MR | Zbl

[15] Gekhtman, M.; Shapiro, M.; Vainshtein, A. Cluster algebras and Poisson geometry, Amer. Math. Soc., Providence, RI, 2010 | MR | Zbl

[16] Glick, M. The pentagram map and Y-patterns (Adv. Math, to appear, arXiv:1005.0598) | MR | Zbl

[17] Henriques, A. A periodicity theorem for the octahedron recurrence, J. Algebraic Combin., Volume 26 (2007) no. 1, pp. 1-26 | DOI | MR | Zbl

[18] Keller, B. The periodicity conjecture for pairs of Dynkin diagrams (arXiv:1001.1880) | MR | Zbl

[19] Marshall, I.; Semenov-Tian-Shansky, M. Poisson groups and differential Galois theory of Schroedinger equation on the circle, Comm. Math. Phys., Volume 284 (2008), pp. 537-552 | DOI | MR | Zbl

[20] Ovsienko, V.; Schwartz, R.; Tabachnikov, S. Liouville-Arnold integrability of the pentagram map on closed polygons (preprint) | MR | Zbl

[21] Ovsienko, V.; Schwartz, R.; Tabachnikov, S. The Pentagram map: a discrete integrable system, Comm. Math. Phys., Volume 299 (2010), pp. 409-446 | DOI | MR | Zbl

[22] Ovsienko, V.; Tabachnikov, S. Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge University Press, Cambridge, 2005 | MR | Zbl

[23] Propp, J. The combinatorics of frieze patterns and Markoff numbers (arXiv:math/0511633)

[24] Schwartz, R. The pentagram map, Experimental Math., Volume 1 (1992), pp. 71-81 | EuDML | MR | Zbl

[25] Schwartz, R. Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., Volume 3 (2008), pp. 379-409 | DOI | MR | Zbl

[26] Scott, J. Grassmannians and cluster algebras, Proc. London Math. Soc., Volume 92 (2006), pp. 345-380 | DOI | MR | Zbl

[27] Soloviev, F. Integrability of the Pentagram Map (arXiv:1106.3950) | MR | Zbl

[28] Tabachnikov, S. Variations on R. Schwartz’s inequality for the Schwarzian derivative (Discr. Comput. Geometry, in print, arXiv:1006.1339) | MR | Zbl

[29] The On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/ njas/sequences)

[30] Volkov, A. On the periodicity conjecture for $Y$-systems, Comm. Math. Phys., Volume 276 (2007), pp. 509-517 | DOI | MR | Zbl