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For $G$ an algebraic group of type $A_l$ over an algebraically closed fieldof 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 explicitdescriptions of the corresponding modules for $s = 3$.

Small-dimensional representations of algebraic groups of type $A_l$