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2Frieze Patterns and the Cluster Structure of the Space of Polygons
"... We study the space of 2frieze patterns generalizing that of the classical CoxeterConway frieze patterns. The geometric realization of this space is the space of ngons (in the projective plane and in 3dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n ..."
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We study the space of 2frieze patterns generalizing that of the classical CoxeterConway frieze patterns. The geometric realization of this space is the space of ngons (in the projective plane and in 3dimensional 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 2frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.
The Pentagram Integrals on Inscribed Polygons
"... The pentagram map is a completely integrable system defined on the moduli space of polygons. The integrals for the system are certain weighted homogeneous polynomials, which come in pairs: E1,O2,E2,O2,... In this paper we prove that Ek = Ok for all k, when these integrals are restricted to the space ..."
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The pentagram map is a completely integrable system defined on the moduli space of polygons. The integrals for the system are certain weighted homogeneous polynomials, which come in pairs: E1,O2,E2,O2,... In this paper we prove that Ek = Ok for all k, when these integrals are restricted to the space of polygons which are inscribed in a conic section. Our proof is essentially a combinatorial analysis of the integrals. 1
GaleRobinson Sequences and Brane Tilings
"... Abastract. We study variants of GaleRobinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using brane tilings, as from the physics literature. Résumé. On étudie des variantes des suites de GaleRo ..."
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Abastract. We study variants of GaleRobinson sequences, as motivated by cluster algebras with principal coefficients. For such cases, we give combinatorial interpretations of cluster variables using brane tilings, as from the physics literature. Résumé. On étudie des variantes des suites de GaleRobinson motivées par les algèbres amassées à coefficients principaux. Pour ces cas, on donne des interprétations combinatoires des variables d’amas en termes de pavages branes, interprétations qui ressemblent à celles qu’on trouve dans des articles de physique.
GLICK’S CONJECTURE ON THE POINT OF COLLAPSE OF AXISALIGNED POLYGONS UNDER THE PENTAGRAM MAPS
"... Abstract. The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axisaligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the po ..."
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Abstract. The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axisaligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the point of collapse is always the center of mass of the axisaligned polygon. In this paper, we answer Glick’s conjecture positively, and generalize the statement to higher and lower dimensional pentagram maps. For the latter map, we define a new system – the mirror pentagram map – and prove a closely related result. In addition, the mirror pentagram map provides a geometric description for the lower dimensional pentagram map, defined algebraically by
ON THE INTEGRABILITY OF THE SHIFT MAP ON TWISTED PENTAGRAM SPIRALS
"... Abstract. In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type (N, 1) posses a Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of tw ..."
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Abstract. In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type (N, 1) posses a Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of twisted spirals, writing the map in terms of the coordinates and associating a natural parameterfree Lax representation. We then show that the map is invariant under the action of a 1parameter group on the moduli space of twisted (N, 1) spirals, which allow us to construct the Lax pair. 1.