  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.
