Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $\rho:\pi_1(S) \rightarrow \mathrm{ PSL}_n(\mathbb{ C})$, there exists a positive representation $\rho_0:\pi_1(S) \rightarrow \mathrm{ PSL}_n(\mathbb{ R})$ that dominates $\rho$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\mathbb{ X}_n= \mathrm{ PSL}_n(\mathbb{ C})/\mathrm{ PSU}(n)$. Moreover, the $\rho_0$-lengths of peripheral curves remain unchanged. The dominating representation $\rho_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.
Categories: math.GT