Reading this interesting short paper review https://haggaim.github.io/projects/universality/poster.pdf, I came across some notations/definitions that I'd like to understand better.
Let $G \le S_n$ any subgroup of the symmetric group $S_n$ and define the action of $G$ on $x \in \mathbb{R}^n$ as
$$g \cdot x := (x_{g^{-1}(1)} \, \dots \, , x_{g^{-1}(n)}) \, \, \, ,g \in G.$$
The action of $G$ on tensors $X \in \mathbb{R}^{n^k \times a}$ is defined as
$$(g \cdot X_{i_1, \dots i_k,j}) = X_{g^{-1}(i_1), \dots, g^{-1}(i_k), j}, \, \, g \in G.$$
Now how should I think about the object $X \in \mathbb{R}^{n^k \times a}$ ..?
if $k=1$ then I would have simply a matrix $X \in \mathbb{R}^{n \times a}$ representing a 2-rank tensor....? But why? I have a bit of confusion about this kind of notation.