Let S be a punctured surface of negative Euler characteristic. We show that given a generic representation \(\rho :\pi _1(S) \rightarrow \textrm{PSL}_n(\mathbb {C})\) , there exists a positive representation \(\rho _0:\pi _1(S) \rightarrow \textrm{PSL}_n(\mathbb {R})\) that dominates \(\rho \) in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space \(\mathbb {X}_n= \textrm{PSL}_n(\mathbb {C})/\textrm{PSU}(n)\) . 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.
