What is the current status of representation theory of $n$-ary groups in terms of hypermatrices? An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I was thinking about how to study $n$-ary groups and realized a lot of our understanding of groups is dependent on the fact that we have a concrete linear representation theory of them. I.E. a group $G$ can be thought of very concretely as collections of matrices, and then abstract questions about the group $G$ become concrete questions in linear algebra/algebraic geometry about the set of matrices, and sometimes that makes the questions tractable.
There is a generalization of the concept of a matrix to a concept called a hyper matrix (it's an $N$-dimensional array of numbers as opposed to just 2 dimensional). And these hypermatrices support a very natural $N$-ary associative multiplicative operation on them (made by splicing the $N$-dimensional arrays of numbers into $N-1$ dimensional subarrays across each index and the taking dot products) , making them a ripe candidate for generalizing linear representation theory to the $n$-ary world.
It appears people know about this nice multiplication property, at least according to the answerer of this post.
But after digging through google and whatever free resources I have I was not able to find any reference in the literature on using hypermatrices to create a representation theory for $n$-ary groups. Just resources about hypermatrices themselves or $n$-ary groups themselves.
Does anyone know if such a project has been conducted before? and if so what are some papers/humans to get a general idea on the status of the field?
Algebraic Definition:
Given 3 arrays of 3-D numbers $A,B,C$ we form their 3-ary product $E_{t,u,v}$ from the sum
$$ \sum_{i,j} A_{t,i,j}B_{i,u,j}C_{i,j,v}$$
Visualization of the multiplication protocol for three 2x2x2 hypermatrices, to compute the first element of their 3-ary product:

 A: Here's one way to invent the representation theory of groups.

*

*Linearize the definition of a group, leading to the definition of an algebra.

*Notice that each group has a corresponding algebra, the group algebra, which perfectly captures its structure.

*Define the representations of a group as the modules over its group algebra.

The first two steps generalize nicely to $n$-ary groups. Maybe a good candidate for a representation of $n$-ary group would be something that generalizes the third step.
Linearizing the definition of an $n$-ary group
An $n$-ary group is a set $G$ with a map $G^n \to G$ satisfying certain properties. You can define an $n$-ary algebra as a vector space $A$ with a linear map $A^{\otimes n} \to A$ satisfying "the same properties." This works because the defining properties can be expressed with string diagrams, which make sense in the category of multilinear maps (or any other monoidal category). For example, here are the string diagram equations that define associativity and identity elements, respectively, in 3-ary groups.


This is the usual definition of an algebra when $n = 2$.
Defining an $n$-ary group algebra.
In the free vector space $\mathbb{C}[G]$ over an $n$-ary group $G$, the group operation extends to a linear map $\mathbb{C}[G]^{\otimes n} \to \mathbb{C}[G]$ with "the same properties." This is the usual definition of the group algebra when $n = 2$.
This answer was very mixed-up in its original form, which proposed that "the elements of an $n$-ary group should be represented by $(n, 1)$-tensors." I've revised it into something more sensible.
A: @Achim Krause was completely correct this operation with cubes is not associative in the sense required for 3-ary groups.
It's still worthwhile to ask what type of associative this cubical product does obey but as far as i'm concerned finding a "good representation theory for 3-ary groups" in terms of rings/fields we are familiar with looks unsolved or at least not well documented at the time of writing this.
