1
vote
0answers
29 views

From linear invariants of group to general ones

There is a lot of information about classical/linear invariants of finite groups. But does it lead to general invariants of group (for example, when we consider some action of our group on finite ...
7
votes
1answer
143 views

On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation ...
2
votes
0answers
39 views

Invariants of parabolic subgroups

Let $G$ be a finite reflection group acting on $R^n$. Then for each point $x\in R^n$ we can look at its stabilizer $G_x$. Since $G$ is a finite reflection group, its ring of invariants is a polynomial ...
1
vote
0answers
27 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
2
votes
0answers
58 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
1
vote
0answers
61 views

$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
2
votes
0answers
35 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...