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

Discrete Euler-Poincaré and Lie-Poisson equations