Past exam of the introduction to representation theory course of the University of Oxford / 2023 / 3 / a

Let $\rho$ afford the character $\chi$, and let $m$ be the order of a group element $g$. Then $\rho (g{)}^m=I$, so its minimal polynomial divides $T^m-1$. Over the complex numbers this polynomial has distinct polynomial roots, since its formal derivative $m{T}^{m-1}$ does not vanish at any of its polynomial roots. Thus $\rho (g)$ is diagonalizable, with eigenvalues ${\lambda }_1,\dots ,{\lambda }_d$ that are roots of unity.

Each ${\lambda }_j$ has absolute value one, so ${\lambda }_j^{-1}=\overline{{\lambda }_j}$. The eigenvalues of $\rho (g^{-1})=\rho (g{)}^{-1}$ are ${\lambda }_j^{-1}$, givingThis is the character inversion identity.