Free Modules of Relative Invariants of Finite Groups

This paper addresses questions about when modules of relative invariants of a finite group G acting on a polynomial ring R are free over the ring of invariant polynomials R^G. A converse (first obtained by Shchvartsman) is proven of a result asserting that these modules are always free when the group is generated by pseudoreflections. We also re-prove the characterization given by Shchvartsman of which characters χ of degree one have the above property, and deduce from this a characterization of which G have the above property for all their degree one characters.