Pentagram maps over rings, Grassmannians, and skewers

TL;DR

This paper develops a unified framework for generalizations of the pentagram map to Grassmannians and skewer geometry by viewing them as pentagram maps over rings, demonstrating their integrability over various rings.

Contribution

It introduces a common algebraic framework for pentagram map generalizations using rings, including non-commutative and non-division rings, and proves their integrability.

Findings

Grassmannian pentagram map corresponds to matrix rings.

Skewer map is a pentagram map over dual numbers.

Pentagram map remains integrable over any stably finite ring.

Abstract

The pentagram map is a discrete dynamical system on planar polygons. By definition, the image of a polygon $P$ under the pentagram map is the polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. The pentagram map was introduced by R. Schwartz in 1992, and is now one of the most renowned discrete integrable systems. Several authors proposed generalizations of the pentagram map to other geometries, in particular to Grassmannians, where the role of points and lines is played by higher-dimensional subspaces, as well to skewer geometry, where both points and lines are affine lines in the three-dimensional Euclidean space. In the present paper, we develop a common framework for these kinds of generalizations. Specifically, we show that those maps can be viewed as pentagram maps in the projective plane over an appropriate ring. In general, those rings need not be division rings or commutative. We show that the Grassmannian pentagram map corresponds to the ring of matrices, while the skewer map is the pentagram map over the ring of dual numbers. Furthermore, we prove that the pentagram map remains integrable for any stably finite ground ring $R$.