Affine root systems, stable tubes and a conjecture by Geiss-Leclerc-Schr\"{o}er

Associated to a symmetrisable Cartan matrix $C$, Geiss-Lerclerc-Schr\"{o}er constructed and studied a class of Iwanaga-Gorenstein algebras $H$. They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of $\tau$-locally free $H$-modules are the positive roots of type $C$ when $C$ is of finite type, and conjectured that this is true for any $C$. In this paper, we look into this conjecture when $C$ is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type $C$ is the rank vector of some $\tau$-locally free $H$-module. However, the converse is not true in general. Our construction shows that there are $\tau$-locally free $H$-modules whose rank vectors are not roots, when $C$ is of type $\widetilde{\mathbb{B}}_n$, $\widetilde{\mathbb{CD}}_n$, $\widetilde{\mathbb{F}}_{41}$ and $\widetilde{\mathbb{G}}_{21}$, and so the conjecture fails in these four types.