  For $G$ an algebraic group of type $A_l$ over an algebraically closed field
of characteristic $p$, we determine all irreducible rational representations of
$G$ in defining characteristic with dimensions $\le (l+1)^s$ for $s = 3, 4$,
provided that $l > 18$, $l > 35$ respectively. We also give explicit
descriptions of the corresponding modules for $s = 3$.
