In this talk, we show that for any generic representation $\rho \colon \pi_1 (S) \rightarrow \text{PSL}_n(\mathbb{C})$, there is a Hitchin representation $j \colon \pi_1 (S) \rightarrow \text{PSL}_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum and the translation length spectrum. Moreover, if $S$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.