Gr\"obner bases for mesh relations and applications to compositions of irreducible morphisms

TL;DR

This paper establishes a necessary condition for the existence of certain compositions of irreducible morphisms in module categories, using Gr"obner bases to analyze mesh relations.

Contribution

It introduces a novel approach applying Gr"obner bases to study mesh relations and morphism compositions in representation theory.

Findings

Provides a criterion linking morphism compositions to zero paths in mesh categories.

Develops Gr"obner bases for subspaces generated by mesh relations.

Enhances understanding of irreducible morphism compositions in algebra.

Abstract

We give a necessary condition for the existence of a path of n irreducible morphisms between indecomposable modules whose composition lies in the (n + 1)-power of the radical. In order to do that, we consider the general criterion given by C. Chaio, P. Le Meur and S. Trepode, which relates these compositions with zero paths in the mesh category, and then study morphisms in the mesh category by providing Gr\"obner bases for the subspaces generated by the mesh relations.