Weakly positive and directed Anosov representations

Given a finitely generated group $\mathrm{\Gamma }$, a directed graph $\mathrm{\Lambda }$, and a map $R:\mathrm{\Lambda }\to \mathrm{\Gamma }$, we introduce the notion of an $(R,\mathrm{\Lambda })$-directed Anosov representation. This is a weakening of the notion of Anosov representations. Our main theorem gives a procedure to construct $(R,\mathrm{\Lambda })$-directed Anosov representations using Fock-Goncharov positivity. As an application of our main theorem, we construct large families of primitive stable representations from $F_2$ to $\mathrm{P}\mathrm{G}\mathrm{L}(V)$, including non-discrete and non-faithful examples.