Finite representation type of a quiver

A quiver has finite representation type over a field when it has only finitely many isomorphism classes of indecomposable finite-dimensional representations.

A connected quiver has finite representation type exactly when its underlying unoriented graph is a Dynkin diagram of type $A$, $D$, or $E$. In that case, dimension vectors give a bijection from indecomposable representations to positive roots.

At a sink $i$, the Bernstein–Gelfand–Ponomarev reflection functor reverses every arrow into $i$ and replaces the vector space at $i$ by the kernel of the sum of its incoming maps. Away from the simple representation supported at $i$, reflection at the resulting source is an inverse functor.