Question

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ acts by permutation on the indeterminates. The subring of invariant polynomials $R^{\mathfrak{S}_n}$ has a nice description (by generators and relations) in terms of symmetric functions.

What if I only consider the action of the cyclic group $Z_n$? Does anyone know if the ring $R^{Z_n}$ admits a nice presentation? (in the case at hand I had $\mathbb{C}[x_1,x_2,x_3]$ and the action of the cyclic group $Z_3$. Maybe in this case we can use some formula (assuming there is one) for groups splitting as semidirect products?)

Answer 1

The actions of $S_n$ and $\mathbb Z_n$ differ in the sense that in the first case the quotient is smooth (it is again $\mathbb C^n$) while in the second case it is singular. This is why in the fist case we have a nice presentation, but in the second not really. For example, the number of generators of the quotient can not be less than the dimension of Zariski tangent space to the singularity at zero of $\mathbb C^n/\mathbb Z_n$.

Still in principle the presentation can be provided by toric geometry (http://www.cs.amherst.edu/~dac/toric.html) because the quotient is the toric singularity. For example, in your case of $\mathbb C^3/\mathbb Z_3$ let us change the coordinates so that $\mathbb Z_3$ is acting as $w_0\to w_0$, $w_1\to \mu w_1$, $w_2\to \mu^2 w_2$ (here $\mu^3=1$). Then you can write the minimal set of four generators:

$w_0, w_1^3, w_2^3, w_1w_2$, and one obvious relation $(w_1^3w_2^3)=(w_1w_2)^3$

The case $\mathbb C^n/\mathbb Z_n$ for $n>3$ will be more involved, but the idea is the same roughly. First you chose the coordinates on $\mathbb C^n$ $w for which the action is diagonal. Then pick the minimal set of monomials (in these new coordinates) that are invariant under the action, and generate the whole set of invariant monomials (of positive degree).

Consider one more case $n=4$, and chose the coordinates $w_i$, so that $Z_4$ is acting as $w_i\to \mu^iw_i$, $\mu^4=1$. The number of generators is $7$ this time:

$w_0, w_1^4, w_3^4, w_2^2, w_1w_3, w_1^2w_2, w_3^2w_2$

Answer 2

By the general theory of invariants of finite groups, there exist $(n-1)!$ homogeneous polynomials $u_1, \dots, u_p$ ($p=(n-1)!$) such that every element $f$ of $R^{Z_n}$ can be uniquely written $f = u_1 g_1 + \cdots+ u_p g_p$, where $g_1,\dots,g_p$ are symmetric functions. I don't know whether an explicit description of $u_1,\dots,u_p$ is known for arbitrary $n$.

Answer 3

This is just to add 1% to Dmitri's 99% complete answer. Change the coordinates to $w_0,\dots, w_{n-1}$ defined by the formula

$$ w_i = x_0 + \mu^i x_1 + \mu^{2i} x_2 + \dots, $$

where $\mu$ is a primitive $n$-th root of identity. Then the ring of invariants is the subring of monomials

$$ w_0^{k_0}\dots w_{n-1}^{k_{n-1}} \quad \text{such that}\quad n\ |\ k_1 + 2k_2 + \dots (n-1) k_n$$

and a set of generators can be obtained by taking minimal such monomials (i.e. not divisible by smaller such monomials). And relations between these generators are of the form (monomial in $w_i$) = (another monomial in $w_i$). That's a pretty easy presentation by any standard.

P.S. This works over $\mathbb C$ or any ring containing $1/n$ and $\mu$.

Answer 4

I recommend that you read the chapter of this book http://www.springerlink.com/content/n7163w70l6472421/ entitled Algorithms and Invariants. In particular it shows that you you can use Groebner bases to calculate generators for the ring of invariants under the action of a finite group.