Components of AR-quivers for string algebras of type C˜ and a conjecture by Geiss-Leclerc-Schröer

We study modules of certain string algebras, which are referred to as of affine type C˜. 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. In particular, we show that an indecomposable module is τ-locally free if and only if it is preprojective, or preinjective or regular in a tube. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type C˜ and τ-locally free modules of the corresponding string algebras. Furthermore, given a positive root α, we show that if α is real, then there is a unique τ-locally free module M (up to isomorphism) with rank_M=α; otherwise there are families of τ-locally free modules with rank_M=α.