0
votes
1answer
47 views

Generating set of the algebra invariants of finite group.

Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of ...
2
votes
2answers
93 views

Is the ring of polynomial invariants of a finite perfect group an UFD?

Let $G$ be a finite group. $G$ acts on $\mathbb K[x_1,...,x_n]$ by automorphisms fixing $K$. $\mathbb K[x_1,...,x_n]^G=\{ T\in \mathbb K[x_1,...,x_n],\forall \sigma \in G, T^{\sigma}=T\}$ is the ring ...
4
votes
2answers
105 views

What is the field of definition of an invariant ideal?

Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings $$ ...
1
vote
1answer
45 views

Spectrum of $\mathbb{C}[x,y]^{\mathbb{C}^*}$

Let $\mathbb{C}[x,y]$ the ring of polynomials with $\mathbb{C}$-coefficients. We can define an action $\phi: \mathbb{C}^* \times \mathbb{C}[x,y] \rightarrow \mathbb{C}[x,y]$ such that ...
1
vote
0answers
27 views

Description of certain invariant polynomials (not a group action)

Working on a recent question led me to the following invariant-computation problem : let $$ A=\bigg\lbrace P \in {\mathbb Q}[X_1,X_2,X_3,X_4] \ \bigg| \\ \quad\ P(X_1X_3+X_2X_4+X_1X_4,\ X_2X_3,\ ...
12
votes
1answer
156 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
3
votes
3answers
136 views

Ring of invariants of Klein Four group

Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial ...
2
votes
1answer
85 views

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field ...
4
votes
2answers
124 views

Finite presentation of algebra of invariants

(1) Let $R$ be a ring, let $A$ be a finitely presented $R$-algebra, and let $G$ be a finite group of $R$-automorphisms of $A$. Is the algebra of invariant $A^G$ finitely presented over $R$? I can ...
3
votes
2answers
182 views

Quotient of an affine variety by a finite group coincides with topological quotient as a point set?

I have just read the construction of the quotient of a closed subset $X$ of affine space by a finite group $G$ of automorphisms of $X$, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ...