The Auslander-Reiten Components in the Rhombic Picture

For an indecomposable module \(M\) over a path algebra of a quiver of type \(\widetilde{\mathbb A}_n\), the Gabriel-Roiter measure gives rise to four new numerical invariants; we call them the multiplicity, and the initial, periodic and final parts. We describe how these invariants for \(M\) and for its dual specify the position of \(M\) in the Auslander-Reiten quiver of the algebra.