Symmetric polynomials on rearrangement-invariant function spaces

The exact representation of symmetric polynomials on Banach spaces with symmetric basis and also on separable rearrangement-invariant function spaces over [0, 1] and [0, infinity) is given. As a consequence of this representation it is obtained that, among these spaces, l(2n), L-2n[0, 1], L-2n[0, infinity) and L-2n[0, infinity) boolean AND L-2n[0, infinity) where n, nl are both integers are the only spaces that admit separating polynomials.