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, explicitly,$$\mathbb{C}[x, y]^G = \{f \in \mathbb{C}[x, y] : f(\zeta^{-1}x, \zeta y) = f(x, y)\}.$$What is the description of the algebra $\mathbb{C}[x, y]^G$ by generators and relations?

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?