Higher pentagram maps, weighted directed networks, and cluster dynamics
By Michael I. Gekhtman, Michael Shapiro, Serge Tabachnikov, A. Vainshtein. 2011
53 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Spectroscopy, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/83f8768d0a2a25d82ed4e318ede49dc318152bf1
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In Geometry and Topology
In Statistical and Nonlinear Physics
Citations in the corpus
- INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, unknown): “OV, AND A. VAINSHTEIN The pentagram map is coming of age, and we finish this introduction by briefly mentioning, in random order, some related work that appeared since our initial research announcement [14] was written. • A variety of multi-dimensional versions of the pentagram map, integrable and non”
- Introduction to Cluster Algebras (2018, background): “pentagram map from the cluster algebra perspective. The first ingredient is a compatible Poisson structure, which can be obtained directly from the Y -pattern analogue of Theorem 5.4. Proposition 6.9 ([42, 23]). Define a Poisson bracket by {yi,yj}= bijyiyj where B is the exchange matrix associated to ”