Quotient of a vector space by a linear finite group action

Question

Let the cyclic group $\mathbb{Z}_n$ act on $\mathbb{C}^n$ (or on $\mathbb{R}^n$, I'm interested in both) by permuting coordinates. What does the topological quotient $Q$ by this group action look like? More explicitly, I'd like to identify points under the equivalence relation $x\sim y\Leftrightarrow x=g\cdot y$ for some $g\in\mathbb{Z}_n$. Is there some nice way to embed it as a subset of $\mathbb{C}^m$ for some $m$?

So far all I've thought of is that if $f: \mathbb{C}\to\mathbb{C}$ is any function, $e_f: \mathbb{C}^n\to\mathbb{C}^n$ is the application of $f$ elementwise, and $\mathcal{F}$ is the discrete Fourier transform, then the map $\mathcal{F}_f^n: \mathbb{C}^n\to\mathbb{C}^n$ defined by $x\mapsto \left(\mathcal{F}\left(e_f(x)\right)\right)^n$ is fixed by the group action and so defines a map out of $Q$. It's easy to see that this map is not injective, but perhaps by concatenating $\mathcal{F}_f^n$ for several different $f$ one can get an injective map and by choosing nicely-behaved $f$ one can get a nice embedding.

But I'd bet there are more illuminating embeddings. I'm happy to consider related questions where some bad points are removed from $\mathbb{C}^n$ (such as multiples of the all-ones vector), $n$ is assumed to be prime, etc. I'm also interested in other (abelian, so far) group actions, like $\mathbb{Z}_n\times\mathbb{Z}_n$ acting on $\mathbb{C}^{n\times n}$, so if there is a general theory of such quotient spaces, I'd be interested to learn about it. It seems like this sort of question must be well-studied, but I am not sure where exactly it fits, so feel free to re-tag.

Answer 1

The action of $\mathbb{Z}_n$ on $\mathbb{C}^n$ can be diagonalized: it's conjugate to the action sending a vector $(z_0, z_1, \dots z_{n-1}) \in \mathbb{C}^n$ to

$$(z_0, \zeta_n z_1, \zeta_n^2 z_2, \dots \zeta_n^{n-1} z_{n-1}).$$

The quotient can be stratified according to which of the $z_i$ are nonzero (starting with $z_1$). On the open stratum where $z_1 \neq 0$, there's a unique representative of each orbit where $0 \le \text{arg}(z_1) < \frac{2\pi}{n}$. In general, on the stratum where $z_1 = \dots = z_{k-1} = 0$ and $z_k \neq 0$, there's a unique representative of each orbit where $0 \le \text{arg}(z_k) < \frac{2 \pi \gcd(k, n)}{n}$. Each stratum lies in the closure of the previous stratum.

There should be a similar and more complicated story for the action of $\mathbb{Z}_n$ on $\mathbb{R}^n$, coming from the decomposition of the latter into real irreducible representations.