Finite covers of graphs, their primitive homology, and representation theory

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{C})$, which is the span of homology classes represented by components of lifts of primitive elements of $\pi_1(X)$. This circle of ideas relates combinatorial group theory, surface topology, and representation theory.