TITLE: Is this quiver with relations of finite representation type
QUESTION [5 upvotes]: Let $Q=(Q_0,Q_1)$ be the following quiver, $Q_0$ consist of 2 vertices, denoted by 1,2. $Q_1$ consist a loop at 1 called $\gamma$, an arrow $\alpha$ from 1 to 2 and an arrow $\beta$ from 2 to 1. The relation $\rho$ is $\{\beta\alpha, \beta\gamma, \gamma\alpha, \gamma^m\}$  for some integer $m\geq 2$. Given a field $k$, is the quiver algebra $kQ$ always of finite representation type?

REPLY [8 votes]: An easy way to see that the algebra is of infinite representation type (for any $m \geq 2$) is to observe that it is a string algebra, and that you have strings of arbitrary length, each corresponding to an indecomposable module.
For instance the strings $(\alpha \beta \gamma^{-1})^t$, $t \geq 1$, correspond to indecomposable modules of vector space dimension $3t+1$.