$\DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\End}{End} $Let $\mathbb K$ be a field and $A$ a finite subgroup of $\GL_n\mathbb K$. For each $M \in \GL_n\mathbb K$, let's define a ring endomorphism $\psi_M \in \End \mathbb K[x_1,\dots,x_n]$ such that \begin{equation} \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \mapsto M \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} . \end{equation}

$f \in \mathbb K[x_1,\dots,x_n]$ is called invariant under $A$ if $\psi_M(f) = f$, for all $M \in A$. The problem is to find the ring of the invariants. For example, for $\mathbb K = \mathbb R$, $n=2$ and $A =\langle -I_2\rangle$, the ring of the invariants is $\mathbb R[x^2,y^2,xy]$. A more complex example can be found here. We can do the same with rational functions instead of polynomials, can't we? I'm not sure the term invariant theory is the correct one, though.

Are there any books or lecture notes about this topic?


You are living in the era of internet and google. Existence question can be easily settled. For finite group invariants and algorithmic oriented treatment see the book by B. Sturmfels or the one by H. Derksen & G. Kemper.

For invariants of linear algebraic groups see the books by Dolgachev, or the one by Procesi. There is one by Mumford Geometric Invariant Theory which revived the subject in 1960s. And the one by Dieudonne & Carrel gives historical account.


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