Gröbner bases for mesh relations and applications to compositions of irreducible morphisms

TL;DR

This work develops Gröbner-basis techniques for mesh categories arising from translation quivers to analyze compositions of irreducible morphisms in representation theory. By introducing mesh-canonical relations and mesh-lexicographic orders, the authors show these relations form a Gröbner basis for the spaces defining mesh categories, enabling the detection of zero paths via reduction. A key result proves that, under closed-path hypotheses, certain composites of irreducible morphisms cannot lie in $\operatorname{rad}^{n+1}$, providing a new necessary condition tied to Auslander–Reiten sequences. The approach connects coverings, mesh categories, and Riedtmann’s functors to yield a general criterion applicable to all $n$ and illuminates structural restrictions on morphism compositions in finite-dimensional algebras.

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 Gröbner bases for the subspaces generated by the mesh relations.

Theorems & Definitions (22)

• Definition 1.1
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 3.1
• Example 3.2
• Definition 4.1
• Definition 4.2
• Remark 4.3
• Proposition 4.4