Orthogonal Determinants of ${\mathrm{G}\mathrm{L}}_n(q)$

Let $n$ be a positive integer and $q$ be a power of an odd prime. We provide explicit formulas for calculating the orthogonal determinants $\det (\chi )$, where $\chi \in \mathrm{I}\mathrm{r}\mathrm{r}({\mathrm{G}\mathrm{L}}_n(q))$ is an orthogonal character of even degree. Moreover, we show that $\det (\chi )$ is “odd”. This confirms a special case of a conjecture by Richard Parker.