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In this article we classify the irreducible representations of finiteclassical groups $G$ of Lie type of rank $\ell$ with dimension at most aconstant proportional to $\ell^4$ according to whether $G$ is of type$A_{\ell}$ or not. The dimensions of those representations are explicitlycomputed by hand. We conclude the work by addressing certain cases of theinverse Galois problem via the foregoing classification.

Classifying representations of finite classical groups of Lie type of dimension up to $\ell^4$