On generalizations of the pentagram map: discretizations of AGD flows

Abstract

In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspaces in $\RP^m$. These maps are natural candidates to generalize the pentagram map, itself defined as the intersection of consecutive shortest diagonals of a convex polygon, and a completely integrable discretization of the Boussinesq equation. We conjecture that the $k$-AGD flow in $m$ dimensions can be discretized using one $k-1$ subspace and $k-1$ different $m-1$-dimensional subspaces of $\RP^m$.