How can I find a composition series of a module of a Hereditary algebra? I have a problem in Representation Theory, I have to find a composition series of an indecomposable module. Consider this quiver $Q$: $$1 \rightarrow 2 \rightarrow 3 \leftarrow 4 \rightarrow 5$$
I have a module $C \in Mod(KQ)$ given by this dimension vector $(01111)$. How can I find a composition series of $C$ in a practical way?
 A: First, you have to find a simple submodule. In case you have a sink in your quiver with a non-zero vector space on it, this is easy, it is just that vector space (as $\operatorname{rad}(KQ)S=0$ for a one-dimensional subspace $S$ of that vector space, here $\operatorname{rad}(KQ)$ denotes the Jacobson radical). If there is no such vertex, then take a vertex neighbouring a sink with a non-zero subspace. A one-dimensional subspace will again be a simple submodule.
I take the quiver $2\to 3\leftarrow 4$ and the dimension vector $(111)$ as an example:


*

*So $3$ is a sink and the obvious submodule with  dimension vector $(010)$ 

*Now if you mod out $(010)$ from $(111)$ you get $(101)$.

*There are two vertices neighbouring a sink with zero subspace, $2$ and $4$. Let's take $2$.

*Then $(100)$ is a simple submodule of $(101)$ and for a composition series you need to take its preimage under the projection map, that is the space with dimension vector $(110)$

*Similarly $(001)\subseteq (001)$ and taking preimages again we get the following composition series:
$$0\subset (010)\subset (110)\subset (111)$$


I did not do your example in order not to spoil the exercise. The procedure is similar. 
