Gregor Kemper
Universitaet Heidelberg


"Symmetric Powers of Modular Representations for Groups with a Sylow
Subgroup of Prime Order" (with Ian Hughes),
J. of Algebra (to appear). 

Abstract:

  Let $V$ be a representation of a finite group $G$ over a field of
  characteristic~$p$. If~$p$ does not divide the group order, then
  Molien's formula gives the Hilbert series of the invariant ring. In
  this paper we find a replacement of Molien's formula which works in
  the case that $|G|$ is divisible by~$p$ but not by~$p^2$. We also
  obtain formulas which give generating functions encoding the
  decompositions of all symmetric powers of $V$ into indecomposables.
  Our methods can be applied to determine the depth of the invariant
  ring without computing any invariants.
  This leads to a proof of a conjecture of the second author on
  certain invariants of $\GL_2(p)$.