Symmetric powers of modular representations,hilbert series and degree bounds

Let G = Z _p be a cyclic group of prime order p with a representation G → GL(V) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring KV]^G. Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Z _p ×H with |H| coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p ² the invariant ring A,KV]G is generated by homogeneous invariants of degrees at most dim (V).(|G| – 1).