Higher pentagram maps, weighted directed networks, and cluster dynamics

Abstract

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras. In this paper, we extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems. Our main tool is Poisson geometry of weighted directed networks on surfaces developed by M. Gekhtman, M. Shapiro, and A. Vainshtein. The ingredients necessary for complete integrability -- invariant Poisson brackets, integrals of motion in involution, Lax representation -- are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter $k>1$. The case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. If $k=2$ and the ground field is $\C$, we give a geometric interpretation in terms of circle patterns.

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References in corpus (2)

• Total positivity, Grassmannians, and networks
• On generalizations of the pentagram map: discretizations of AGD flows

Cited by in corpus (5)

• Lie Groups, Cluster Variables and Integrable Systems
• Y-meshes and generalized pentagram maps
• Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups
• Glick's conjecture on the point of collapse of axis-aligned polygons under the pentagram maps
• The cluster variety face of quantum groups