Degrees of irreducible morphisms and finite-representation type

TL;DR

This paper characterizes the degrees of irreducible morphisms in Auslander-Reiten components and establishes a criterion for finite representation type of algebras based on these degrees.

Contribution

It provides a new characterization of finite representation type using degrees of specific morphisms and generalizes previous results on compositions of irreducible morphisms.

Findings

Finite representation type is characterized by finite degrees of certain morphisms.

Provides criteria involving radicals and socles for algebra classification.

Generalizes Igusa and Todorov's results on sectional paths.

Abstract

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. We also apply the techniques that we develop: We study when the non-zero composite of a path of $n$ irreducible morphisms between indecomposable modules lies in the $n+1$-th power of the radical; and we study the same problem for sums of such paths when they are sectional, thus proving a generalisation of a pioneer result of Igusa and Todorov on the composite of a sectional path.