So I was studying the local shape (around the simple representation $S(3)$) of the Auslander Reiten quiver of this $Q$:
And so far I concluded that the neighborhood of the representation $S(3)$ looks like this:
Where the representations $S(3)$ and $\begin{matrix} 2 \\3\end{matrix}$ on the right and left side are identified. (so the AR quiver actually looks like a tube locally)
I wanted to prove that the three representations $N_1$, $N_2$ and $N_3$ are not isomorphic. Is this obvious from the graph, or how can I show it without messing too much with explicit calculations?
So far I proved that if two of them are isomorphic, all of them are isomorphic. And that implies (because of the uniqueness of the almost split sequences), that all the three representations in the second row, must have arrows (irreducible morphisms) into all three $N_i$'s. So there is one almost split sequence (one for each $N_i$ actually) $0 \rightarrow N \rightarrow \begin{matrix} 1 \\ 2 \ 4\end{matrix} \oplus \begin{matrix} 1 \ 3 \\ 4\end{matrix} \oplus \begin{matrix} 2 \\ 3 \end{matrix} \rightarrow N \rightarrow 0$.
But I don't see any contradiction. Is there any way to prove that they are not isomorphic by looking at the AR quiver or some abstract nonsense?
I would use that the socle functor is left exact. This way you can see that $S(3)\in \operatorname{soc} N_1$ by applying the socle functor to the Auslander-Reiten sequence starting in $\begin{smallmatrix}2\\3\end{smallmatrix}$ and similarly, one can see that $S(3)\notin \operatorname{soc} N_2$ by applying the socle functor to the Auslander-Reiten sequence starting in $\begin{smallmatrix}&1\\2&&4\end{smallmatrix}$.