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In this paper, we study representations of certain string algebras, which arereferred to as of affine type $\widetilde{C}$. We introduce minimal stringmodules and apply them to explicitly describe components of theAuslander-Reiten quivers of the string algebras and $\tau$-locally free modulesdefined by Geiss-Lerclerc-Schr\"{o}er. As an application, we proveGeiss-Leclerc-Schr\"{o}er's conjecture on the correspondence between positiveroots of type $\widetilde{C}$ and $\tau$-locally free modules of thecorresponding string algebras.

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