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)? $