Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$.
Can this theorem be generalized into higher dimensions? That is, are there any simple restrictions to put on a set of (finite-dimensional) representations of a given (semi?) group $G$ over a fixed (algebraically closed?) field $K$ so as to assure that their characters are linearly independent? It is natural to assume that the representations are irreducible (otherwise, obviously the character of $\pi$ and $\pi\oplus \pi$ are linearly dependent, and the latter would even turn out to be $0$ if $\operatorname{char} K=2$), and in case of $K=\bf C$ and finite group $G$ I suppose irreducibility is enough by Schur's orthogonality (so I guess this is also true for algebraically closed $K$ of characteristic $0$ or large enough for a given $G$ by some model-theoretical argument).
This question arose out of curiosity about the theorem as stated in a commutative algebra course, and I have little to no idea about modular representation theory, or even any non-$\bf C$ representation theory.
Summing it all up, is there a known theorem that generalizes Artin's theorem, and if not, is there any reason that there isn't (perhaps the reason being that it is trivial from some viewpoint?)?