Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying:
$$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, n\}$$
Is it possible to extend this "root of unity property", for lack of a better term, to some special value in a system with more than two values? For example, is there some hypercomplex number system, or matrix perhaps, that somehow admits one or more values that have this property? My understanding is that this may be possible in a ring.
WHAT I'M AFTER
Essentially, I'm trying to find a system with more than 2 components that has something that functions like a principle root of unity.
One possible generalization is to the non-abelain case. It is the well-known Schur's orthogonality theorem on irreducible characters of a finite group.