Action of a group $G$ on copies of $G$ Let $G$ be a group and $G^n$ be the direct product of $n$ copies of $G$. There is an action $\psi$ of $G$ on the set $G^n$ such that :
$\psi\left(g,(g_1,\;g_2,\ldots,\;g_n)\right) = \left(g_1 \cdot g \cdot g_1^{-1} \cdot g_1, \;g_1 \cdot g \cdot g_1^{-1} \cdot g_2, \ldots, \;g_1 \cdot g \cdot g_1^{-1} \cdot g_n\right)$
I'm studying this from scratch, but I'm sure this group action has already appeared somewhere. My question is therefore: has this kind of action already been studied, and if so, what was the context ?
 A: Your action is more standard after relabeling the domain. Possibly you are dealing with more of a "projective" space, and chose the "wrong" way to make it look affine.
Consider the map $f:G^n \to G^n$ given by $f(g_1,g_2,\dots,g_n) = (g_1,g_1 g_2, \dots, g_1 g_n)$.
Then $f^{-1}:G^n \to G^n$ is given by $f^{-1}(g_1,g_2,\dots,g_n) = (g_1, g_1^{-1} g_2, \dots, g_1^{-1} g_n)$ and $\psi^f : G \to \operatorname{Sym}(G^n)$ is a right action given by:
$$\begin{align*}
\psi^f \left(g, \left(g_1,g_2,\dots,g_n \right)\right)
&= f^{-1}\left( \psi\left( g, f\left(g_1,g_2,\dots,g_n\right)\right)\right) \\
&= f^{-1}\left( \psi\left( g, \left(g_1,g_1 g_2, \dots, g_1 g_n\right)\right)\right) \\
&= f^{-1}\left( g_1g, g_1g g_2, \dots, g_1g g_n \right) \\
&= \left( g_1 g, g^{-1} g_2, \dots, g^{-1} g_n \right)
\end{align*}
$$
This is more or less the standard right action on $G^n$. In the first coordinate it is right-multiplication by $g$, and on the other coordinates it is left-multiplication by $g^{-1}$. Each is standard, but I guess it is not horribly common to use both at the same time.
