Semi-Invariants for Gentle String Algebras

TL;DR

This paper presents an algorithm to determine generators and relations for semi-invariant rings of gentle string algebras, using matching graphs to analyze their structure and degree bounds, applicable to acyclic string algebras.

Contribution

It introduces a novel graph-based method to compute semi-invariant rings and their relations for gentle string algebras, extending to acyclic cases.

Findings

Generators correspond to certain walks on the matching graph

Relations are characterized by specific graph configurations

Degree bounds are established for generators and relations

Abstract

In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of representation spaces for gentle string algebras. These rings of semi-invariants turn out to be semigroup rings to which we can associate a so-called matching graph. Under this association, generators for the semigroup can be seen by certain walks on this graph, and relations are given by certain configurations in the graph. This allows us to determine degree bounds for the generators and relations of these rings. We furthermore show that these bounds also hold for acyclic string algebras in general.