Higher pentagram maps, weighted directednetworks, and cluster dynamics

Michael Gekhtman; Michael Shapiro; Serge Tabachnikov; Alek Vainshtein

doi:10.3934/era.2012.19.1

Article Contents

2012, Volume 19: 1-17. Doi: 10.3934/era.2012.19.1

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Higher pentagram maps, weighted directed networks, and cluster dynamics

1.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
2.
Department of Mathematics, Michigan State University, East Lansing, MI 48823
3.
Department of Mathematics, Penn State University, University Park, PA 16802
4.
Department of Mathematics and Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905

Received: September 30, 2011

Revised: November 30, 2011

Published: December 2011

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Abstract

The pentagram map was extensively studied in a series of papers by V. Ovsienko, R. Schwartz and S. Tabachnikov. It was recently interpreted by M. Glick as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of geometrically meaningful discrete integrable maps.

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Mathematics Subject Classification: Primary: 13F60; Secondary: 53D17.

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