The pentagram map on Grassmannians

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.