Mesh-comparable components of the Auslander-Reiten quiver
Viktor Chust, Flávio U. Coelho
TL;DR
The paper develops the notion of mesh-comparable components of the Auslander-Reiten quiver, defined via a Riedtmann functor $F:k(\Gamma)\to\mathrm{ind}\,\Gamma$, to study compositions of irreducible morphisms without passing to coverings. It introduces a decomposition of morphisms into parts (finite and infinite) along the radical filtration, and defines principal/secondary parts and depth to analyze morphism structure. A central result shows that generalized standard mesh-comparable components are standard, while standard components are mesh-comparable; the framework also yields a practical criterion for when compositions of chosen morphisms land in higher radical layers and connects these properties to irreducible-morphism degrees. Altogether, the work unifies mesh-comparability, standardness, and morphism decomposition to advance understanding of morphism composition in AR components with direct, computable implications. The approach provides two new proofs that standard implies generalized standard and offers tools for degree computations via chosen morphisms, with implications for both theoretical and computational representation theory.
Abstract
The idea of using Riedtmann's well-behaved functors to study compositions of irreducible morphisms has been explored in a number of articles. Here we introduce the concept of mesh-comparable components of the Auslander-Reiten quiver, which are components for which a Riedtmann functor exists without the necessity of taking a covering, such as the universal or the generic one. We show properties of this type of component, and study the problem of compositions of irreducible morphisms in this context.