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

In this paper, we study representations of certain string algebras, which are referred to as of affine type C̃. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and τ -locally free modules defined by Geiss-Lerclerc-Schröer. As an application, we prove Geiss-Leclerc-Schröer’s conjecture on the correspondence between positive roots of type C̃ and τ -locally free modules of the corresponding string algebras.