Y-meshes and generalized pentagram maps

Abstract

We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$-mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry. Our framework incorporates many preexisting generalized pentagram maps due to M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein and also B. Khesin and F. Soloviev. In several of these cases a reduction to cluster dynamics was not previously known.

arXiv comment48 pages, 22 figures, to appear in Proceedings of the London Mathematical Society

References in corpus (7)

• Gauge Theories from Toric Geometry and Brane Tilings
• Cluster mutation-periodic quivers and associated Laurent sequences
• Liouville-Arnold integrability of the pentagram map on closed polygons
• Integrability of the Pentagram Map
• Non-integrability vs. integrability in pentagram maps
• Higher pentagram maps, weighted directed networks, and cluster dynamics
• Brane Tilings and Non-Commutative Geometry

Cited by in corpus (5)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Vector-relation configurations and plabic graphs
• The Schwarzian octahedron recurrence (dSKP equation) II: geometric systems
• Discrete dynamics in cluster integrable systems from geometric $R$-matrix transformations
• Lagrangian configurations and symplectic cross-ratios