  For $X$ a curve over a field of positive characteristic, we investigate when
the canonical representation of $\text{Aut}(X)$ on $H^0(X, \Omega_X)$ is
irreducible. Any curve with an irreducible canonical representation must either
be superspecial or ordinary. Having a small automorphism group is an
obstruction to having irreducible canonical representation; with this
motivation, the bulk of the paper is spent bounding the size of automorphism
groups of superspecial and ordinary curves. After proving that all
automorphisms of an $\mathbb{F}_{q^2}$-maximal curve are defined over
$\mathbb{F}_{q^2}$, we find all superspecial curves with $g > 82$ having an
irreducible representation. In the ordinary case, we provide a bound on the
size of the automorphism group of an ordinary curve that improves on a result
of Nakajima.
