The authors consider the pentagram map, a discrete dynamical system on the space of planar polygons introduced by R. Schwartz [Exp. Math. 1, No. 1, 71–81 (1992; Zbl 0765.52004)]. It turns out that this is a discrete integrable system. The authors prove integrability of long-diagonal pentagram maps on polygons in \(\mathbb{R}P^d\) and study an equivalence of long-diagonal and bi-diagonal maps. A self-contained construction in the frame of Lax theory is presented. It turns out that the pentagram map admits different integrable generalizations to multidimensional spaces. The bi-diagonal map associated with progressions \(A_{\pm }\) is defined as follows.
Denote by \(A_{+},A_{-}\subset \mathbb{Z}\) two disjoint finite non-empty arithmetic progressions with the same common difference, each containing at least two elements. Then the bi-diagonal map associated with the stated progressions is a self-map of the space of polygons in \(\mathbb{R}P^d\) defined in a special way. For instance, the standard pentagram map in \(\mathbb{R}P^2\) is a bi-diagonal map corresponding to \(A_{+} = \{0, 2\}, A_{-} = \{1, 3\}\).
The first interesting result in the paper is that all long-diagonal pentagram maps are completely integrable discrete dynamical systems on the space of projective equivalence classes of twisted \(n\)-gons in \(\mathbb{R}P^d\). Each of these maps admits a Lax representation with spectral parameter and an invariant Poisson structure such that the spectral invariants of the Lax operator Poisson commute. The second result is the description of the continuous limit for long-diagonal maps. The statement is that the continuous limit of any long-diagonal pentagram map in dimension \(d\) is the \((2, d + 1)\)-KdV flow of the Adler-Gelfand-Dickey hierarchy on the circle.