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.
Highlights
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
Schwartz [21] showed that the continuous limit of the pentagram map is the Boussineq equation
The first author [9] proved that the pentagram map can be described in certain coordinates as mutations in a cluster algebra
Summary
By definition of a Y -mesh, the points Pr+a+c, Pr+b+c are distinct and both lie on the line Lr+c. An (L2) relation shifted by b shows Pr+2b, Pr+b+c, Pr+b+d are collinear. Starting from A ∈ X2,S, we can iterate T2,S and T2−,S1 to fill up the whole Pi,j array, and all (L1) and (L2) relations will hold It follows that generically Pr+a, Pr+b, Pr+c, and Pr+d are collinear. By a (P3) relation, the four points used in the construction are coplanar making it possible to join them in pairs and intersect the resulting lines. From the above and an (L1) relation, the points Pr+a+b, Pr+2b, Pr+b+c, Pr+b+d generically are distinct and lie on a line.
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