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 theoryInvariant theory, Chapter V oof Bourbaki's Lie groups and Lie algebrasLie groups and Lie algebras (englishEnglish translation), and Benson's book Polynomial invariants of finite groupsPolynomial invariants of finite groups.
