Let $\mathrm{SL}_n/K$ ($K$ finite) be given with its natural action on an $n$-dimensional vector space $V/K$. Consider the action of $\mathrm{SL}_n$ on the $m$-fold tensor product $V\otimes \dotsc \otimes V$. What is the largest-dimensional irreducible subrepresentation?

(For $m$ much larger than the maximum of $n$ and the size of $K$, it cannot be the symmetric power, as its dimension becomes larger than the size of $\mathrm{SL}_n(K)$. Or am I missing something?)