Every representations of $A_n$ of degree $n^{O(1)}$ is contained in a $O(1)$ tensor power of the defining permutation representation. Is there an analogous result for classical groups of Lie type, say $G = \text{SL}_n(q)$ ($q$ bounded, $n$ large)? The defining action of $G$ on $F_q^n$ gives rise to a complex (permutation) representation $V$ of degree $q^n-1$, and the dual action likewise a representation $W$ also of degree $q^n-1$. Is it true that every representation of degree $q^{O(n)}$ is contained in $V^{\otimes O(1)} \otimes W^{\otimes O(1)}$?

Every representation of $A_n$ of degree $n^{O(1)}$ is contained in a $O(1)$ tensor power of the defining permutation representation. Is there an analogous result for classical groups of Lie type, say $G = \text{SL}_n(q)$ ($q$ bounded, $n$ large)? The defining action of $G$ on $F_q^n$ gives rise to a complex (permutation) representation $V$ of degree $q^n-1$, and the dual action likewise a representation $W$ also of degree $q^n-1$. Is it true that every representation of degree $q^{O(n)}$ is contained in $V^{\otimes O(1)} \otimes W^{\otimes O(1)}$?