## Background

I have a set $S$ (that is possibly infinite) and a correspondence between functions $c:S^3tomathbb{R}$ (I will write $c(i,j,k)$ as $c_{ijk}$) and matrices $M$ with rows indexed by $(i,j,k)in S^3$ and columns indexed by $(m,n)in S^2$ defined as follows:

$$M_{(i,j,k),(m,n)}=delta_{im}c_{njk}+delta_{jm}c_{nik}-delta_{ln}c_{ijm}$$

where $delta_{ij}$ is the kronecker delta function.

## Question

**I want to determine some necessary and sufficient conditions on $c$ so that the above matrix has trivial null space.**

Any references, ideas, techniques, etc. would be appreciated. Of course, I’m not looking for a complete solution to the problem, I just don’t know what sort of methods might even be used to attack this (despite it appearing at first to be a simple linear algebra problem).

## Progress

For finite $S$ with cardinality $N$, this is a $N^3times N^2$ size matrix, so it *should* be true that most choices of $c$ result in a matrix with trivial null space. However, I’m still not sure what “most” would even mean in this context.

I’ve also proven the result for $c_{ijk}=delta_{ijk}$ (that is $c_{iii}=1$ and equals zero otherwise), but otherwise have no further results.