  In this paper, discrete analogues of Euler-Poincar\'{e} and Lie-Poisson
reduction theory are developed for systems on finite dimensional Lie groups $G$
with Lagrangians $L:TG \to {\mathbb R}$ that are $G$-invariant. These discrete
equations provide ``reduced'' numerical algorithms which manifestly preserve
the symplectic structure. The manifold $G \times G$ is used as an approximation
of $TG$, and a discrete Langragian ${\mathbb L}:G \times G \to {\mathbb R}$ is
construced in such a way that the $G$-invariance property is preserved.
Reduction by $G$ results in new ``variational'' principle for the reduced
Lagrangian $\ell:G \to {\mathbb R}$, and provides the discrete
Euler-Poincar\'{e} (DEP) equations. Reconstruction of these equations recovers
the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are
naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP
algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is
shown that when $G=\text{SO} (n)$, the DEP and DLP algorithms for a particular
choice of the discrete Lagrangian ${\mathbb L}$ are equivalent to the
Moser-Veselov scheme for the generalized rigid body. %As an application, a
reduced symplectic integrator for two dimensional %hydrodynamics is constructed
using the SU$(n)$ approximation to the volume %preserving diffeomorphism group
of ${\mathbb T}^2$.
