Inverse systems and regular representations

Let G be a finite group acting on a finite-dimensional vector space V, such that the ring of invariants is polynomial. The purpose of this note is to describe exactly the finitely generated inverse systems such that the associated G-representation is the direct sum of copies of the regular representation of G. This generalizes work of Steinberg, Bergeron, Garsia, and Tesler. Related results are also recalled. All of the results are contained in the main theorem.