Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by algebra automorphisms. The celebrated Chevalley-Shephard-Todd theorem says that $P_n^G$ is polynomial if and only if $G$ is a pseudo-reflection group.

However, there are many more conditions which characterize when the invariant subalgebra $P_n^G$ is polynomial. I'm looking for an exaustive list of such known conditions.

Below there are the equivalent conditions I know

Theorem: Let $k$, $G$ and $P_n$ be as above. Then the following are equivalent

i) $P_n^G$ is a polynomial algebra (necessarily in $n$ indeterminates)

ii) $G$ is a pseudo-reflection group

iii) $P_n^G$ is a regular ring

iv) $P_n$ is a finitely generated free $P_n^G$-module

v) $P_n$ is a finitely generated projective $P_n^G$-module

vi) $P_n$ is a finitely generated flat $P_n^G$-module

vii) The map $P_{n+}^G \otimes_{P_n^G} P_n \rightarrow P_n$ given by multiplication is injective ($P_{n+}^G$ is the ideal of $P_n^G$ generated by elements of positive degree).

The equivalence of all the above conditions can be found in a combination of T. A. Springer's book on invariant theory, Chapter V o Bourbaki's Lie groups and Lie algebras (english translation), and Benson's book Polynomial invariants of finite groups.