  We investigate the relationship between multipartite entanglement and
symmetry, focusing on permutation symmetric states. We use the Majorana
representation, where these states correspond to points on a sphere. Symmetry
of the representation under rotation is equivalent to symmetry of the states
under products of local unitaries. The geometric measure of entanglement is
thus phrased entirely as a geometric optimisation, and a condition for the
equivalence of entanglement measures written in terms of point symmetries.
Finally we see that different symmetries of the states correspond to different
types of entanglement with respect to SLOCC interconvertibility.
