Affine Root Systems, Stable Tubes, and a Conjecture by Geiss-Leclerc-Schröer

Associated to a symmetrizable Cartan matrix, Geiss-Leclerc-Schröer constructed and studied a class of Iwanaga-Gorenstein algebras. They proved a generalized version of Gabriel's Theorem, that is, the rank vectors of-locally free-modules are the positive roots of type when is of finite type, and conjectured that this is true for any. In this paper, we investigate this conjecture specifically for the case when is of affine type. We explicitly construct stable tubes, some of which are inhomogeneous tubes with non-rigid mouth modules. This characteristic is not presented in the representation theory of quivers (with no relations). We deduce that any positive root of type is the rank vector of some-locally free-module. However, the converse is not generally true. Our construction shows that there are-locally free-modules whose rank vectors are not roots, when is of type, and, and so the conjecture fails for these four types.