The study of entanglement between more than two systems, or particles, is complicated by the fact that the number of degrees of freedom of n-particles, each with a d-dimensional state space, increases steeply as a function of n an d. By requiring symmetry both under particle permutation and under joint unitary transformations, we simplify the situation considerably and yet retain a rich, fundamental structure, which might serve as a kind of “backbone” for the general theory. We lean heavily on the classical representation theory of finite and compact groups. The well-known Young diagrams occurring in the representation theory of finite permutation groups are seen in a new light. A connection is made with Lieb’s conjecture concerning the so-called immanants of positive definite matrices. A promising approach to characterize the set of all separable (non-entangled) states as a polytope in state space, breaks down in a strange way at n=5.