Classify the finite-dimensional indecomposable representations of the quiver $A_n$

Question

I have to classify the finite-dimensional indecomposable representations of the quiver $A_n$ up to isomorphism:

1 $\longrightarrow$ 2 $\longrightarrow \cdots \longrightarrow$ n

Any idea of how to approach this??

Answer 1

There is a general theory which implies that the indecomposable representations of a quiver that arises as an orientation of a simply-laced Dynkin diagram of finite type are in bijection with the positive roots of the corresponding root system, in such a way that the dimension vector of an indecomposable representation is given by the coefficients of the corresponding positive root on the simple roots. This result is due to Gabriel, with a later important refinement of Bernstein, Gelfand, and Ponomarev. The main tool in this general approach is the set of reflection functors defined by BGP.

However, in your case the algebra of the quiver is simply the algebra of $n$ by $n$ upper triangular matrices, and you may approach the problem via that identification.