I solved the following problem using Molien's theorem, but I would like to know another solution.
Let $R = \oplus R_n$ be the subring of the graded ring $\mathbb{C}[x,y]$ of polynomials invariant under the automorphism that multiplies $x$ and $y$ by $e^{2\pi i/k}$, where $k$ is a positive integer. Calculate the function $\sum_n \dim(R_n)z^n $, where $R_n$ is the subspace of polynomials in $R$ of degree $n$.
Thanks in advance!
Here is my solution:
Using Molien's theorem, we get:
$\sum_{n = 0}^{\infty}(\dim R_n)z^n = \frac{1}{k} \sum_{j = 0}^{k - 1} \frac{1}{\det(I_2 - ze^{2\pi i/j}I_2)} = \frac{1}{k} \sum_{j = 0}^{k - 1} \frac{1}{(1 - ze^{2\pi i/j})^2} = \frac{1}{k} \sum_{j = 0}^{k - 1} \sum_{n = 1}^{\infty} n(ze^{2\pi i/j})^{n - 1} = \frac{1}{k} \sum_{n = 1}^{\infty}\bigg( \sum_{j = 0}^{k - 1} e^{2\pi i (n - 1)/j} \bigg) n z^{n - 1} = \frac{1}{k} \sum_{n = 0}^{\infty} (n + 1)\big(\sum_{j=0}^{k-1} e^{2 \pi n i/j}\big)z^n$
Also, I would like to know how to transform the last expression since I need integers as coefficients of $z^n$ (they represent the dimension of $R_n$).