Let $G$ be an affine algebraic group (i.e. a $k$-variety which is also a group and the group multiplication and inversion are morphisms of varieties). A character of $G$ is a morphism of algebraic groups $\chi:G\to k^\times$.
You may assume for the following that $k^\times$ acts algebraically on $G$, but I do not know if that is required at all.
It is well-known that I can always embed $G$ as a closed, algebraic subgroup of some $\mathrm{Gl}_n$. Now given any (nonzero) character $\chi$, can I embed $G$ in such a way that $\chi=\det_n|_G$, the restriction to $G$ of the determinant on $\mathrm{Gl}_n$? If no, can you characterize the characters that do satisfy this property? Because certainly, some characters of $G$ arise in this manner.