Representation-Infinite Quiver

[Tsakanikas Nickos]

Let \( Q \) be a quiver. Show that if \( Q \) contains a subquiver of the type \( \tilde{A}_{n} \) which is not acyclic, then \( Q \) is representation-infinite, i.e. there are infinitely many isomorphism classes of indecomposable representations of \( Q \).