Difference Operators and Pentagram Maps Over Rings

Abstract

The pentagram map, first proposed by R. Schwartz in 1992, is a well studied discrete integrable system on real planar polygons. It has been reframed in the language of difference operators using a known correspondence of certain degree 3 difference operators and polygons in P2. In this paper, we generalize the notion of a projective space by describing one-dimensional “subspaces” in free modules over stably finite rings. We then define polygons in such spaces and discuss how these projective spaces relate to known objects, such as Grassmannians. With these projective spaces, we generalize the correspondence of difference operators and polygons by proving that there is a one-to-one correspondence between properly bounded left (resp. right) difference operators, whose coefficients are from a stably finite ring R, and polygons in a certain left (resp. right) projective space over R. With this correspondence, we show that some known (and proposed) generalizations of the pentagram map can be understood as the usual pentagram map in a certain projective plane. Finally, we reframe these pentagram maps in the language of difference operators, showing that the pentagram map on pseudo-difference operators is a refactorization, and use this interpretation to find invariants.