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

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Abstract Associated to a symmetrizable Cartan matrix $C$, Geiss–Leclerc–Schröer constructed and studied a class of Iwanaga–Gorenstein algebras $H$. They proved a generalized 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 investigate this conjecture specifically for the case when $C$ 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 $C$ is the rank vector of some $\tau $-locally free $H$-module. However, the converse is not generally true. Our construction shows that there are $\tau $-locally free $H$-modules whose rank vectors are not roots, when $C$ is of type $\operatorname{\widetilde{\mathbb{B}}}_{n}$, $\operatorname{\widetilde{\mathbb{C}\mathbb{D}}}_{n}$, $\operatorname{\widetilde{\mathbb{F}}}_{41}$ and $\operatorname{\widetilde{\mathbb{G}}}_{21}$, and so the conjecture fails for these four types.