[Espace des 2-frises, espace de polygones et structure de variété amassée]
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 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 -gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus curves with marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.
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.
@article{AIF_2012__62_3_937_0,
author = {Morier-Genoud, Sophie and Ovsienko, Valentin and Tabachnikov, Serge},
title = {2-frieze patterns and the cluster structure of the space of polygons},
journal = {Annales de l'Institut Fourier},
pages = {937--987},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {62},
number = {3},
year = {2012},
doi = {10.5802/aif.2713},
mrnumber = {3013813},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2713/}
}
TY - JOUR AU - Morier-Genoud, Sophie AU - Ovsienko, Valentin AU - Tabachnikov, Serge TI - 2-frieze patterns and the cluster structure of the space of polygons JO - Annales de l'Institut Fourier PY - 2012 SP - 937 EP - 987 VL - 62 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2713/ DO - 10.5802/aif.2713 LA - en ID - AIF_2012__62_3_937_0 ER -
%0 Journal Article %A Morier-Genoud, Sophie %A Ovsienko, Valentin %A Tabachnikov, Serge %T 2-frieze patterns and the cluster structure of the space of polygons %J Annales de l'Institut Fourier %D 2012 %P 937-987 %V 62 %N 3 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2713/ %R 10.5802/aif.2713 %G en %F AIF_2012__62_3_937_0
Morier-Genoud, Sophie; Ovsienko, Valentin; Tabachnikov, Serge. 2-frieze patterns and the cluster structure of the space of polygons. Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 937-987. doi : 10.5802/aif.2713. http://archive.numdam.org/articles/10.5802/aif.2713/
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