If $G$ is a finite group, $g \in G$, and $\chi$ is a character of $G$, what are the definitions of "$\chi(g)$" and "$\overline{\chi(g)}$"?

Question

I'm looking at lemma $2.15$ on page $20$. https://www.cefns.nau.edu/~falk/classes/511/Isaacs_Character_theory.pdf

Answer 1

$\chi(g)$ is the trace of the linear transformation $\rho(g)$. Since $G$ is finite, $g^n = 1$ for some $n$ ( works for $n = |G|$). Therefore $\rho(g)^n = I$ and so all of the eigenvalues of $\rho(g)$ are $n$-th roots of $1$. Now, $\chi(g)$ is the sum of these eigenvalues. From before, any eigenvalue $\lambda$ of $1$ has absolute value $1$, so $\bar \lambda = \lambda^{-1}$. Therefore $$\overline{\chi(g)} = \overline {\sum \lambda_l} = \sum \overline {\lambda_l} = \sum \lambda_l^{-1} = \chi(g^{-1})$$