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 τ-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 τ-locally free H-module. However, the converse is not true in general. Our construction shows that there are τ-locally free H-modules whose rank vectors are not roots, when C is of type Bₙ, CDₙ, F₄₁ and G₂₁, and so the conjecture fails in these four types.