Y-meshes and generalized pentagram maps

TL;DR

This paper introduces a broad family of generalized pentagram maps called Y-meshes, linking cluster algebra mutations with projective geometry, and unifies many existing generalizations under this new framework.

Contribution

It establishes a new framework connecting cluster algebra mutations with generalized pentagram maps, including many previously unconnected cases.

Findings

Introduces Y-meshes as a new class of pentagram maps.

Shows all systems have an infinite configuration of points and lines.

Connects cluster algebra theory with projective geometry.

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.