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

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-dimesional arrays of numbers into (N-1) dimensional sub arrays 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 this guy.

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?