Let $\mathbf{k}$ be a field of characteristic 0 and $S=\mathbf{k}[X_1,\cdots,X_n]$ be a polynomial algebra over $\mathbf{k}$. Let $G\subset GL_n(\mathbf{k})$ act linearly on $S$ and $R = S^G$ be the ring of invariants.
We know that $R$ is f.g. $\mathbf{k}$-algebra, say $R=\mathbf{k}[f_1,\ldots, f_r]$.
Using results from Smith's book, we know that $f_1,\ldots,f_r$ is a regular sequence if and only if $S$ is a free $R$-module, of rank $|G|$. Some related questions are here and here.
My question is: What can we say about $S$ as an $R$-module otherwise? i.e. if $f_i$'s do not form a regular sequence?
- Is $S$ finitely generated?
- If yes, is there any way to compute generators of $S$ as $R$-module?
- Any bound on the number of generators?
- Any known results on this module structure? -- may be in spacial cases?
From computational point of view, is there any CAS that can do this computation?
Please correct me if I am wrong: Possible examples of such groups I can think of are the ones which generate $A,D,E$ singularities.
