How to actually find almost split sequences?

Question

I was trying to construct the AR-quiver, or at least a part of it, for the quiver $$Q = \quad\begin{array}{ccccc} & & 1 & & \\ & ^\delta \swarrow & & \searrow ^\alpha & \\ 4 & & & & 2. \\ & _\gamma\nwarrow & & \swarrow_\beta & \\ & & 3 & & \end{array}$$ So I know that $ \tau S(2)=S(3)$ so there is an almost split sequence $0 \rightarrow S(3) \rightarrow E \rightarrow S(2) \rightarrow 0$. So I know that the indecomposable modules located inbetween $S(3)$ and $S(2)$ are the direct summands of $E$. The problem is that I simply don't know any way to calculate any (the only) almost split sequence between $S(3)$ and $S(2)$. I know it exists, because of theory. And I know that is unique up to isomorphism. I also know some tricks for when the quiver is a Nakayama algebra, or when one of the extremes is projective-injective. Otherwise I don't know how could I actually find an almost split sequence between the two. Is there any general method? Any idea?

Answer 1

So the only way to do this I found is manually: comparing the dimension of the vector spaces at each vertex and using linear algebra, $0 \rightarrow 3 \rightarrow \begin{matrix} 2 \\ 3 \end{matrix} \rightarrow 2 \rightarrow 0$ is the only exact sequence that makes sense, it doesn't split, and you can manually prove that the inclusion $3 \rightarrow \begin{matrix} 2 \\ 3 \end{matrix}$ is left minimal almost split using the definition of L.M.A.S. and proving manually point by point that it satisfies the definition (is not a section, factors each other morphism that is not a section and so on).

So I don't know any piece of "theory" that allows me to generalize or prove a sequence is almost split for harder representations.