  Motivated by recent experimental progress in the context of ultra-cold
multi-color fermionic atoms in optical lattices, we have developed a method to
exactly diagonalize the Heisenberg $SU(N)$ Hamiltonian with several particles
per site living in a fully symmetric or antisymmetric representation of
$SU(N)$. The method, based on the use of standard Young tableaux, takes
advantage of the full $SU(N)$ symmetry, allowing one to work directly in each
irreducible representations of the global $SU(N)$ group. Since the $SU(N)$
singlet sector is often much smaller than the full Hilbert space, this enables
one to reach much larger system sizes than with conventional exact
diagonalizations. The method is applied to the study of Heisenberg chains in
the symmetric representation with two and three particles per site up to $N=10$
and up to 20 sites. For the length scales accessible to this approach, all
systems except the Haldane chain ($SU(2)$ with two particles per site) appear
to be gapless, and the central charge and scaling dimensions extracted from the
results are consistent with a critical behaviour in the $SU(N)$ level $k$
Wess-Zumino-Witten universality class, where $k$ is the number of particles per
site. These results point to the existence of a cross-over between this
universality class and the asymptotic low-energy behavior with a gapped
spectrum or a critical behavior in the $SU(N)$ level $1$ WZW universality
class.
