  We introduce the principal representation category $\mathscr{O}({\bf G})$ of
reductive algebraic groups with Frobenius maps and put forward a conjecture
that this category is a highest weight category. When $\Bbbk$ is complex field
$\mathbb{C}$, we provide some evidences of this conjecture. We also study
certain kind of bound quiver algebras whose representations are related to the
principal representation category $\mathscr{O}({\bf G})$ .
