# Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant

$\mathbf{Question \;1}$. Let $$A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right).$$ Then $C= \langle A\rangle$ is a cyclic, finite group of order $4$.

Suppose $A$ acts on $\mathbb{C}[x,y]$ linearly.

Then what is the subring $\mathbb{C}[x,y]^C$ of invariant functions in $\mathbb{C}[x,y]$? What is the basic strategy?

Note that $$C = \left\{ \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right), \left( \begin{array}{cc} -1 & 0 \\ 0& -1 \\ \end{array} \right), \left( \begin{array}{cc} 0 & 1 \\ -1& 0 \\ \end{array} \right), \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) \right\}.$$

$\mathbf{Question \;2}$. Now, suppose the dihedral group $D_6 = \langle \rho, \psi : \rho^6 = \psi^2 =e,\psi \rho\psi^{-1}=\rho^{-1} \rangle$ acts on $\mathbb{C}[x,y,z]$, with the action defined by the matrices $$\rho = \left( \begin{array}{ccc} 1/2 & -\sqrt{3}/2 & 0 \\ -\sqrt{3}/2 & 1/2 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) \mbox{ and } \psi = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & -1 \\ 0 & 0 & -1 \\ \end{array}\right).$$

Then what is $\mathbb{C}[x,y,z]^{D_6}$?

$\mathbf{Question \;3}$. What is the general strategy, if we have something like the subgroup generated by $B$ and $-B$ in $GL_3(\mathbb{C})$ acting on a polynomial ring $\mathbb{C}[x,y]$ of only two variables, where $$B = \left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & -1 \\ 0 & 0& 1 \\ \end{array} \right)?$$

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The basic strategy I'm aware of is to write down some invariant polynomials and then use Molien's theorem to check that you've found a set of generators. –  Qiaochu Yuan Jul 14 '12 at 6:27
I agree with Qiaochu. There is a very nice book which treats this material: Mara Neusel's Invariant Theory. See e.g. amazon.com/Invariant-Theory-Student-Mathematical-Library/dp/… –  Pete L. Clark Jul 14 '12 at 6:31
@QiaochuYuan: Thanks for the reply! –  math-visitor Jul 14 '12 at 6:35
@PeteL.Clark Thanks for the reply! I studied group theory but I just realized that I've never worked with finite groups acting on a polynomial ring before. That's why I'm asking these questions. And I just learned how to "ping" people, and I think you're supposed to ping one person at a time. –  math-visitor Jul 14 '12 at 6:36