Am 16.08.2011 03:06, schrieb Nils Bruin:
> 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.
In my case S only contains monomes like \prod x_{i}^{a_i}
> 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?
Yes, maybe this is the best way to say it.
> 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?
>
The only extra condition I see from my data (but I'm not sure if it
holds every time), is that the degree of all polynomes is not relatively
primely and the polynomes wich degree is the gcd of all degrees arise in S.
greatz Johannes
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