# Stability and Indecomposability of the Representations of Quivers of $A_n$-type

## TL;DR

This paper proves Markus Reineke's conjecture for $A_n$-type quivers by constructing a stable weight system and relates it to semi-invariant theory, advancing understanding of quiver representations.

## Contribution

It provides a combinatorial construction of a stable weight system for $A_n$-type quivers and links it to semi-invariant theory, confirming Reineke's conjecture in this case.

## Key findings

- Confirmed Reineke's conjecture for $A_n$-type quivers.
- Developed a combinatorial method for constructing stable weight systems.
- Reinterpreted the weight system using semi-invariant theory.

## Abstract

In his paper \cite{MR1}, Markus Reineke proposed a conjecture that there exists a stable weight system $\Theta$ for every indecomposable representation of Dynkin type quiver. In this paper, we showed this conjecture is true for quivers of $A_n$-type by combinatorial construction of a special weight system. We also reinterpret this weight system in terms of semi-invariant theory.