Twisted reductions of integrable lattice equations, and their Lax representations

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg-de Vries equation, discrete modified Korteweg-de Vries equation and the discrete Schwarzian Korteweg-de Vries equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.