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