The Pentagram map on Grassmannians

TL;DR

This paper generalizes the pentagram map to Grassmannian spaces, introduces invariants for twisted polygons, and constructs a Lax representation to facilitate integrability analysis.

Contribution

It extends the pentagram map to Grassmannians, defines new invariants, and develops a Lax representation for integrability.

Findings

The pentagram map preserves a specific scaling in Grassmannian coordinates.

Invariants of Grassmannian twisted polygons are identified and used as moduli space coordinates.

A Lax representation for the generalized pentagram map is constructed, enabling integrability studies.

Abstract

In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n;mn). We define invariants of Grassmannian twisted polygons under the natural action of SL(nm), invariants that define coordinates in the moduli space of twisted polygons. We then prove that when written in terms of the moduli space coordinates, the pentagram map is preserved by a certain scaling. The scaling is then used to construct a Lax representation for the map that can be used for integration.