Leaps in the depth of compositions of irreducible morphisms

TL;DR

This paper constructs specific algebra examples demonstrating that compositions of irreducible morphisms can reach arbitrarily high depths within the radical layers, challenging previous assumptions about their limitations.

Contribution

It introduces a family of algebras with controlled compositions of irreducible morphisms, illustrating new possible depths of these compositions in representation theory.

Findings

Existence of algebras with morphism compositions reaching arbitrary radical depths

Construction of string and representation-finite algebras with these properties

Examples showing the depth of compositions can exceed previous bounds

Abstract

In this article, we give a family of examples of algebras, showing that for every $n \geq 2$ and $m \geq 0$, there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the $(n+m+3)$-th power of the radical, but not in the $(n + m + 4)$-th power. Such an algebra may be also supposed to be string and representation-finite.