Reading the following poster https://haggaim.github.io/projects/universality/poster.pdf
A higher order tensor is defined as $X \in \mathbb{R}^{n^k \times a}$ and given a group $G$, its action on $X$ is defined as
$$(g \cdot X)_{i_1, \dots, i_k,j} = X_{g^{-1}(i_1), \dots g^{-1}(i_k),j}$$
The term $a$ is not actually defined exactly, but in light of the context I just think it could refer to the set of all tensors of a certain rank $k$. For example if we take $k =2$, then we have
$$X \in \mathbb{R}^{n \times n \times a}$$ Is it correct to interpret this as the stack of all possible $n \times n$ matrices?
Also, $j$ above is used to describe the feature depth, but I don't really understand why couoldn't we just leave $i_k$ as final index.