The symmetric , D - invariant and Egorov reductions of the quadrilateral lattice

We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, 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. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic and independent reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the d-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and d-invariant. 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.