The Auslander-Reiten quivers of string algebras of affine type $\widetilde{C}$ and a conjecture by Geiss-Leclerc-Schr\"{o}er

TL;DR

This paper characterizes the Auslander-Reiten quivers of affine type or string algebras, introduces minimal string modules, and proves a conjecture linking positive roots to  modules.

Contribution

It provides an explicit description of Auslander-Reiten components for affine type rom string algebras and confirms a conjecture relating roots to modules.

Findings

Explicit description of Auslander-Reiten quivers for affine type rom string algebras

Introduction of minimal string modules for these algebras

Proof of the Geiss-Leclerc-Schrf6er conjecture connecting roots and modules

Abstract

In this paper, we study representations of certain string algebras, which are referred to as of affine type $\widetilde{C}$. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and $\tau$-locally free modules defined by Geiss-Lerclerc-Schr\"{o}er. As an application, we prove Geiss-Leclerc-Schr\"{o}er's conjecture on the correspondence between positive roots of type $\widetilde{C}$ and $\tau$-locally free modules of the corresponding string algebras.