  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 $U_q(g)$.
