Description of the algebra of $G$-invariant polynomials by generators and relations

Question

Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = \text{diag}(\zeta, \zeta^{-1})$. The group $G$ acts naturally on $\mathbb{C}[x, y]$. Let $\mathbb{C}[x, y]^G$ be the algebra of $G$-invariant polynomials, namely $$\mathbb{C}[x, y]^G = \{f \in \mathbb{C}[x, y] : f(\zeta^{-1}x, \zeta y) = f(x, y)\}.$$What is the description of the algebra $\mathbb{C}[x, y]^G$ in terms of generators and relations?

Answer 1

Since the group $G$ has the order $n$ then by Noether's upper bound theorem for generator of the algebra invariants is a polynomial of degree $ \leq n.$ Let us consider the Reynolds average operator $R: \mathbb{C}[x,y] \to \mathbb{C}[x,y]^G$: $$ R=\frac{1}{n}\sum_{g \in G} g. $$

Then, by direct calculation we get
$R((xy)^k)=(xy)^k, k \in \mathbb{N}$ and $R(x^k)=\delta_{k,n} x^k$ and $R(y^k)=\delta_{k,n} y^k.$

So, $\mathbb{C}[x,y]^G=\mathbb{C}[x^n,y^n,xy].$ The relation $x^n \cdot y^n=(xy)^n.$