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

EDIT. This is a new version of the answer. Since at some point there was confusion between the $(\zeta, \, \zeta)$-action and the $(\zeta^{-1}, \, \zeta)$-action, for the sake of clarity let me discuss both.

The $(\zeta^{-1}, \, \zeta)$-action (the OP case).

In this case there are three invariant monomials, namely $$T_0:= x^n, \quad T_1=xy, \quad T_2=y^n.$$ Then the invariant algebra $\mathbb{C}[x, \, y]^G$ is isomorphic to $$\mathbb{C}[T_0, \, T_1, \, T_2]/(T_0T_2-T_1^n),$$ which is is a Rational Double Point (= Du Val singularity) of type $A_{n-1}$, whose resolution graph is the Dynkin diagram with the same label, corresponding to a chain of $n-1$ smooth rational curves with self-intersection $(-2)$. This is a Gorenstein singularity for all $n \geq 2$, for instance because it is a hypersurface singularity, or because $G \subset \textrm{SL}_2(\mathbb{C})$.

The $(\zeta, \, \zeta)$-action.

In this case the quotient $\mathbb{C}^2/G$ is isomorphic to the affine cone over a rational normal curve $C_n \subset \mathbb{P}^n$.

Therefore $\mathbb{C}[x, \, y]^G$ is the ring of regular functions of such a cone, namely the ring $\mathbb{C}[T_0, \ldots, T_n]/I$, where $I$ is the ideal generated by the $2 \times 2$ minors of the matrix $$\begin{equation} \begin{pmatrix} T_0 & T_1 & \ldots & T_{n-1}\\ T_1 & T_2 & \ldots & T_n \end{pmatrix}. \end{equation}$$

Here the variables $T_i$ correspond to the invariant monomials for the action, namely $T_i:=x^{n-i}y^i$. The singularity at the vertex of the cone is of type $\frac{1}{n}(1, \, 1)$, has embedding dimension $n+1$ and is not Gorenstein for $n \geq 3.$ The resolution is a unique smooth rational curve with self-intersection $(-n)$.

Remark.

The two cases coincide for $n=2$, when the action is simply given by $(x, \, y) \to (-x,\, -y)$. The invariant algebra is generated by the three invariant polynomials $$T_0:=x^2, \quad T_1:=xy, \quad T_2:=y^2$$ subject to the unique relation $T_0T_2-T_1^2=0,$ which is the affine cone over a smooth conic in $\mathbb{P}^2$.