Components of AR-quivers for string algebras of type (C)over-tilde and a conjecture by Geiss-Leclerc-Schroer

We study modules of certain string algebras, which are referred to as of affine type (C) over tilde. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and t-locally free modules defined by Geiss-Lerclerc-Schroer. In particular, we show that an indecomposable module is tlocally free if and only if it is preprojective, or preinjective or regular in a tube. As an application, we prove GeissLeclerc-Schroer's conjecture on the correspondence between positive roots of type (C) over tilde and tau-locally free modules of the corresponding string algebras. Furthermore, given a positive root alpha, we show that if ais real, then there is a unique tau-locally free module M(up to isomorphism) with rankM= alpha; otherwise there are families of tau-locally free modules with rankM=alpha. (c) 2023 Elsevier Inc. All rights reserved.