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?



Is this quiver with relations of finite representation type
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](http://www.math.uni-bonn.de/people/schroer/fd-atlas-files/FD-BiserialAlgebras.pdf), 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 $3t+1$.


