Invariants cyclic group actions

Question

Let $G_m$ be the multiplicative group of $m$-th roots of unity generated by $\epsilon_m=\exp{(2\pi i/m)}$, and let us assume it acts faithfully on $\mathbb{C}[x_1,x_2]$ with weights $a=(a_1,a_2)=(a_1,m-a_1)$, where $0<a_1<m$, that is, $\epsilon(x_i)=\epsilon^{a_i}\cdot x_i$.

Find generators of $\mathbb{C}[x_1,x_2]^{G_m}$ and find relations between generators.

I have just started a basic course on invariant theory, but I've already encountered many problems, as I don't have any idea how to solve this problem.

1. Find generators of $\mathbb{C}[x_1,x_2]^{G_m}$: is it true that, in this case, $$\mathbb{C}[x_1,x_2]^{G_m}=\{f\in \mathbb{C}[x_1,x_2]\mid f(x_1,x_2)= f(\epsilon ^{a_1}x_1,\epsilon^{m-a_1}x_2) \}?$$ So, intuitively, I should check when $f(x_1,f_2)=f(\epsilon ^{a_1}x_1,\epsilon^{m-a_1}x_2)$. I tried to consider the basic case $m=2\rightarrow a_1=1$, thus the map is the identity and $\mathbb{C}[x_1,x_2]^{G_2}=\mathbb{C}[x,y]$. I tried to consider $m=3\rightarrow a_1=1$ or $a_1=2$, thus we have two identities to consider: $f(x_1,x_2)=f(\epsilon x_1, \epsilon^2 x_2)$ and $f(x_1,x_2)=(\epsilon^2 x_1,\epsilon x_2)$. How should I proceed now? I mean, I should equalize term per term, but the "$\epsilon$" must be treat as coefficient or variable? How to deal the general $m$-case?
2. Find relations of $\mathbb{C}[x_1,x_2]^{G_m}$: I really don't understand what that means.

I'm sorry, I know it's evident I 've no idea idea what I'm doing, but I'd like to understand. Any help would be much appreciate, thanks in advance.

Answer 1

Let $d=\gcd(a_1,m)$ then $(\epsilon_m^{a_1}, \epsilon_m^{m-a_1}) = (\epsilon_n^{b_1}, \epsilon_n^{n-b_1})$ where $m = nd$ and $a_1 = b_1 d$. Hence we may assume that $a_1$ and $m$ are coprime. Now the main idea is to workout the action on monomials. This will lead to some easy congruence equations that caracterize the invariant monomials hence the invariant polynomials. This solves Part 1. For Part 2 the relations will be obvious once you know the generators.

Bellow the full answer:

Since the action of $G_m$ preserves monomials we look for the invariant monomials first:$$ \epsilon_m \cdot x^py^q = \epsilon_m^{a_1p+ (m-a_1)q}x^py^q = \epsilon_m^{a_1(p-q)}x^py^q $$ Therefore $x^py^q$ is invariant if and only if $m$ divides $a_1(p-q)$ and since $\gcd(a_1,m)=1$ it is equivalent to $$p \equiv q \pmod{m}$$ The we have the following monomials of minimal degree $\{ x^m, xy, y^m\}$. It is easy to prove that any other invariant monomial must be of the form $x^{rm}(xy)^sy^{tm}$ for some integers $r,s,t$. It follows that $$\mathbb{C}[x,y]^{G_m} = \mathbb{C}[x^m,xy, y^m]. $$For part 2) just note that $(xy)^m = x^m y^m$ hence $\mathbb{C}[x^m,xy, y^m] \simeq \dfrac{\mathbb{C}[X,Y,Z]}{(Y^m-XZ)}$.