Primary invariants of a finite group

Question

For a finite group $G$ and complex representation V of degree $n$, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants qualify to be called primary ? Or does the set need to satisfy additional conditions ?

I would like to know specifically the following : For a set $f_1,f_2,\ldots,f_n$ of algebraically independent homogeneous invariants, is the full ring of invariants of G a free module over the subalgebra generated by $f_i$'s.

What I have done:For the group $S_5$ and its irreducible representation of degree 6 (This is not generated by reflections), I have found six homogeneous invariants . I also computed the degrees of them whose product is a multiple of the order of $S_5$.

Answer 1

You should trust Max Horn. Definition 2.4.6 in our book has the finiteness condition explicitly. (This is the numbering in the first edition of the book.) For finite groups, this is equivalent to saying that the variety given by the primaries consists only of the origin (see Propos. 3.3.1).

A counter example arises if you take K[V] = K[x,y] with the trivial group action and choose $f_1=x$ and $f_2 = x y$. These are algebraically independent, but you need all the powers of $y$ to generate $K[V]$ as a module over $K[f_1,f_2]$.

If you have primary invariants in the correct sense, the invariant ring is a finitely generated free module over $K[f_1,\ldots,f_n]$ by Cohen-Macauly property. In the above counter example, the module is neither free nor finitely generated.