On Aug 15, 2:54 pm, Johannes <dajo.m...@web.de> wrote:
> I'm sorry for unclear description of the problem.
> So once again, let R = C[x_1,\dots,x_n]$ be my basering.
> I'm looking for the group G, wich leaves a finite set S of polynomes
> invariant under its action. So the ideal I = <S> is invariant under the
> G-action too. And because every constant polynome is invariant under the
> action, I can look at the subring C[I] = C[S] \subset R instead of
> looking at I.
These are not the same rings, though. If S={1} then I=R, so C[I]=R and
C[S]=C.
For your original question, do you mean: Let G subset GL(n,C) be a
matrix group, consider the polynomial ring R=C[x1,...,xn] and let
I=R^G be the G-invariant subring of R. What is the minimal subgroup H
subset G such that
R^H=R^G?
I don't think that question would be well-posed in general, since
there could be several non-conjugate subgroups H1,H2 of G with
R^H1=R^H2=R^G, such that for no proper subgroup H3 of H1 or H2 we have
R^H3=R^G. Perhaps your setting has some properties that guarantee a
unique minimal one? Perhaps those extra properties help in determining
it?
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