  We describe a framework for the controllability analysis of networks of $n$
quantum systems of an arbitrary dimension $d$, {\it qudits}, with dynamics
determined by Hamiltonians that are invariant under the permutation group
$S_n$. Because of the symmetry, the underlying Hilbert space, ${\cal
H}=(\mathbb{C}^d)^{\otimes n}$, splits into invariant subspaces for the Lie
algebra of $S_n$-invariant elements in $u(d^n)$, denoted here by
$u^{S_n}(d^n)$. The dynamical Lie algebra ${\cal L}$, which determines the
controllability properties of the system, is a Lie subalgebra of such a Lie
algebra $u^{S_n}(d^n)$. If ${\cal L}$ acts as $su\left( \dim(V) \right)$ on
each of the invariant subspaces $V$, the system is called {\it subspace
controllable}. Our approach is based on recognizing that such a splitting of
the Hilbert space ${\cal H}$ coincides with the {\it Clebsch-Gordan} splitting
of $(\mathbb{C}^d)^{\otimes n}$ into {\it irreducible representations} of
$su(d)$. In this view, $u^{S_n}(d^n)$, is the direct sum of certain $su(n_j)$
for some $n_j$'s we shall specify, and its {\it center} which is the Abelian
(Lie) algebra generated by the {\it Casimir operators}. Generalizing the
situation previously considered in the literature, we consider dynamics with
arbitrary local simultaneous control on the qudits and a symmetric two body
interaction. Most of the results presented are for general $n$ and $d$ but we
recast previous results on $n$ qubits in this new general framework and provide
a complete treatment and proof of subspace controllability for the new case of
$n=3$, $d=3$, that is, {\it three qutrits}.
