When is the character of a representation a character?

Question

Let $G$ be a group with a complex, finite dimensional representation $r:G \to \operatorname{GL}(V)$.

What is the condition to make $\operatorname{trace}(r(g))$ also a character of $G$, i.e., group homomorphism?

Answer 1

$r(e)=id$ so $trace (r(e))= \dim V$, so it cannot be a group morphism unless $\dim V =1$, in which case it is a morphism

Answer 2

Let $\mathfrak{X}$ be an irreducible $\mathbb{C}$-representation of $G$ affording the character $\chi$. Assume that $\chi(g)\chi(h)=\chi(gh)$ for all $g,h \in G$ (so $\chi$ is a homomorphism). Then in particular $\chi(g)\chi(g^{-1})=\chi(g)\overline{\chi(g)}=|\chi(g)|^2=\chi(1)$. But $1=\frac{1}{|G|}\sum_{g \in G}|\chi(g)|^2=\frac{1}{|G|}|G|\cdot\chi(1)$, hence $\chi$ is linear.