Finite Dimensional Representations of Quantum Affine Algebras

We investigate the characters of some finite-dimensional representations of the quantum affine algebras $U_q(\hat{g})$ using the action of the copy of $U_q(g)$ embedded in it. First, we present an efficient algorithm for computing the Kirillov-Reshetikhin conjectured formula for these characters when $g$ is simply-laced. This replaces the original formulation, in terms of "rigged configurations", with one based on polygonal paths in the Weyl chamber. It also gives a new algorithm for decomposing a tensor product of any number of representations of $sl(n)$ corresponding to rectangular Young diagrams, in a way symmetric in all the factors. This section is an expanded version of q-alg/9611032 . Second, we study a generalization of certain remarkable quadratic relations that hold among characters of $sl(n)$ (the "discrete Hirota relations") whose solutions seem to be characters of quantum affine algebras. We use show that these relations have a unique solution over characters of $U_q(g)$.