Chevalley–Shephard–Todd theorem

In the case of subgroups of the complex general linear group the theorem was first proved by G. C. Shephard and J.

It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre.

Let V be a finite-dimensional vector space over a field K and let G be a finite subgroup of the general linear group GL(V).

There has been much work on the question of when a reductive algebraic group acting on a vector space has a polynomial ring of invariants.

In the case when the algebraic group is simple all cases when the invariant ring is polynomial have been classified by Schwarz (1978) In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so it is a finite-rank free module over a polynomial subring.