Gregor Kemper
Universitaet Heidelberg

"Symmetric Powers of Modular Representations, Hilbert Series and
Degree Bounds",
Comm. in Algebra 28 (2000), 2059-2088.

Abstract:

  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
  K[V]^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 x
  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 K[V]^G is generated by homogeneous invariants of degrees at
  most dim(V) (|G|-1).