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.
Categories: math.RT math.RA