The Auslander-Reiten Components in the Rhombic Picture

TL;DR

This paper introduces four new numerical invariants derived from the Gabriel-Roiter measure for modules over quivers of type e4a4a4a4, and explains how these invariants determine the modules' positions in the Auslander-Reiten quiver.

Contribution

It defines four novel invariants from the Gabriel-Roiter measure and links them to the modules' positions in the Auslander-Reiten quiver for type e4a4a4a4 quivers.

Findings

Four new numerical invariants introduced.

Invariants determine module positions in the Auslander-Reiten quiver.

Application to modules over e4a4a4a4 quivers.

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.