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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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    @PeteL.Cla$r$$k$ Than$k$s fo$r$ the reply! I $s$tudied group theory but I just realized that I've never worked with finite groups acting on a polyno$m$ial ri$n$g before. That's why I'm asking these questions. An$d$ I just lea$r$ned how to "ping" people, and I think you're $s$uppose$d$ to pi$n$g one person at a ti$m$e.2012-07-14

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