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We introduce a notion of representation for a class of generalised quiversknown as Coxeter quivers. These representations are built using fusioncategories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unityand we show that many of the classical results on representations of quiverscan be generalised to this setting. Namely, we prove a generalised Gabriel'stheorem for Coxeter quivers that encompasses all Coxeter--Dynkin diagrams --including the non-crystallographic types $H$ and $I$. Moreover, a similarrelation between reflection functors and Coxeter theory is used to show thatthe indecomposable representations correspond bijectively to the positive rootsof Coxeter root systems over fusion rings.

Coxeter quiver representations in fusion categories and Gabriel's theorem