Symmetric powers, indecomposables and representation stability

35 pages. Comments welcome; Working over the prime field F_p, the structure of the indecomposables Q^* for the action of the algebra of Steenrod reduced powers A(p) on the symmetric power functors S^* is studied by exploiting the theory of strict polynomial functors. In particular, working at the prime 2, representation stability is exhibited for certain related functors, leading to a Conjectural representation stability description of quotients of Q^* arising from the polynomial filtration of symmetric powers.