Thin representations for quiver algebras

Question

A representation $M$ of a quiver is called thin when $M$ has a dimension vector consisting only of 0 or 1 entries. When $A=kQ$ is a path algebra for a tree $Q$, then there is the nice result that indecomposable thin representations correspond to connected subquivers of $Q$.

Question 1: Is there a more general result for (acyclic if needed) quiver algebra $A=KQ/I$ that gives a classification of thin representations? When are there only finitely many thin indecomposable representations?

Question 2: Is there a quick way to obtain all thin indecomposable representations of a given quiver algebra using QPA (maybe just for path algebra $kQ$, even when $Q$ is a tree)?

Answer 1

Answer to Question 1.

Given a thin indecomposable representation $X$ of $k Q$, consider the subquiver $Q'$ consisting of the vertices $i$ with $X_i$ nonzero and the arrows $i\to j$ with the linear map $X_i\to X_j$ nonzero. Clearly $Q'$ is a connected subquiver of $Q$ (but not necessarily full).

Let $T$ be a spanning tree for $Q'$. Then we can choose bases so that $X$ is isomorphic to a representation with a copy of $k$ at each vertex in $Q'$ and arrows in $T$ given by the identity map. The remaining arrows are uniquely determined elements of the multiplicative group $k^* = k\setminus\{0\}$.

Thus the isomorphism classes of thin indecomposable representations of $k Q$ corresponding to $Q'$ are indexed by $(k^*)^n$ where $n$ is one plus the number of arrow in $Q'$ minus the number of vertices in $Q'$.

It follows that $k Q$ has only finitely many thin indecomposable representations if and only if $Q$ is a disconnected union of trees.