Dominating surface-group representations via Fock-Goncharov coordinates

Let S be a punctured surface of negative Euler characteristic. We show that given a generic representation: ₁ (S) PSLₙ (C), there exists a positive representation ₀: ₁ (S) PSLₙ (R) that dominates in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space Xₙ= PSLₙ (C) /PSU (n). Moreover, the ₀-lengths of peripheral curves remain unchanged. The dominating representation ₀ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.