Consider the quiver $1 \overset{\alpha}{\underset{\alpha'}{\leftleftarrows}} 2$. I am trying to find the form of the Auslander–Reiten quiver. So I got :
$P(1)= \begin{matrix} 1 \end{matrix}$, $P(2) =\begin{matrix} 2 &\\ 1 & 1 \end{matrix}$, $I(1) =\begin{matrix} 2 & 2\\ 1 & \end{matrix}$, $I(2)= \begin{matrix} 2 \end{matrix}$, $S(1)=1$, $S(2)=2$. I am just curious about how the Auslander–Reiten quiver looks like.
The result is too long to post here so I just give a reference. There are two approaches to find the Auslander-Reiten quiver (over an algebraically closed field): One more or less just linear algebra based (and thus elementary) approach is described in the book of Barot: Introduction to the Representation theory of algebras Chapter 6.2. As a special result about the representation-theory of tame hereditary algebras, the Auslander-Reiten quiver is computed very fast and nice in the Book of Simson and Skowronski: Elements of the Representation Theory of Associative Algebras: Volume 2. If you dont want to assume that the field is algebraically closed, then Chapter 4.5 in the first volume of the books by Benson helps you and gives a relation to the indecomposable representations of the klein four group over a field of characteristic 2.