In Karpilovsky's Group Representations (vol. 1) it can be read in chapter 27, that if $\chi$ is a character of $G$, and $\rho \colon G \to \mathrm{GL}_{n}(\mathbb{C})$ is a representation which affords $\chi$, then the determinant of $\chi$, defined for any $g\in G$ as
$$ (\det \chi)(g) = \det (\rho(g)) $$
is a linear character that does not depend upon a choice of $\rho$.
Can someone explain this to me? Why will the value of $(\det \chi)(g)$ be the same no matter the representation? Also, why will we have $(\det(\chi))(gh) = (\det \chi)(g) (\det \chi)(h)$ for multiplicativity? Thank you very much in advance!
Best Answer
For any two choices $\rho$ and $\rho'$ giving the same character $\chi$, there exists $P\in GL_n(\mathbb{C})$ such that $\rho'(g)=P\rho(g)P^{-1}$ for all $g\in G$: in other words, $\rho$ and $\rho'$ are equivalent.
Then $\det(\rho'(g))=\det(P\rho(g)P^{-1})=\det(\rho(g))$.
It is multiplicative simply because both $\rho$ and $\det$ are: $\det(\rho(gh))=\det(\rho(g)\rho(h))=\det(\rho(h))\det(\rho(h))$.