The Auslander-Reiten Components in the Rhombic Picture

Abstract

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.Comment: 29 pages; 6 figures; references added to Section 1 as per the referee's suggestions; to appear in Communications in Algebr