The symmetric and Egorov reductions of the quadrilateral lattice

We present a detailed study of the basic reductions of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net). To make this study, it is necessary to introduce new important ingredients in the by now well established theory of quadrilateral lattices. In particular, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the τ–function of the lattice, defined as the potential connecting these data, and to introduce the notion of symmetric lattice, for which forward and backward rotation coefficients coincide. Combining the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) with the symmetric one, we obtain the Egorov lattice, for which we present several interesting characterizations. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a ∂̄ formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.