  We consider a quantum system of n qudits and the Clebsch-Gordan decomposition
of the associated Hilbert space. In this decomposition, one of the subspaces is
the so-called symmetric subspace or symmetric sector, that is, the subspace of
all states that are invariant under the action of the symmetric group. We prove
that any separable state must have a nonzero component on the symmetric sector,
or, equivalently, any state which has zero component on the symmetric sector
must be entangled. For the cases of n=2,3 particles, and in arbitrary dimension
d, this result can be refined by providing sharp lower bounds on the size of
the component of separable states on the symmetric sector. This leads us to
identify a class of entanglement witnesses for these systems. We provide an
example showing that in the multipartite case, this class of witnesses detects
PPT entangled states.
