  Building on advanced results on permutations, we show that it is possible to
construct, for each irreducible representation of SU(N), an orthonormal basis
labelled by the set of {\it standard Young tableaux} in which the matrix of the
Heisenberg SU(N) model (the quantum permutation of N-color objects) takes an
explicit and extremely simple form. Since the relative dimension of the full
Hilbert space to that of the singlet space on $n$ sites increases very fast
with N, this formulation allows to extend exact diagonalizations of finite
clusters to much larger values of N than accessible so far. Using this method,
we show that, on the square lattice, there is long-range color order for SU(5),
spontaneous dimerization for SU(8), and evidence in favor of a quantum liquid
for SU(10).
