Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems

Using Galois covering theory and equivariant techniques, we classify all connected representation-tame Geiß–Leclerc–Schröer (GLS) algebras in terms of their defining triples $(C,D,\mathrm{\Omega })$. We then study the GLS algebras $H({\widetilde{CD}}_n)$ with minimal symmetrizers, where $n\ge 2$ and ${\widetilde{CD}}_2={\tilde{B}}_2$. By realizing these algebras as basic algebras of ${\mathbb{Z}}_2$-skew group algebras of the string algebras $H({\tilde{C}}_{2n-2})$, we introduce extended strings and extended bands to parameterize the connected components of the Auslander–Reiten quivers of $H({\widetilde{CD}}_n)$ and determine their shapes. We further classify the indecomposable $\tau$-locally free modules over representation-tame GLS algebras of affine type and show that their rank vectors form precisely the set of positive roots together with explicitly described non-root vectors in the positive cone of the root lattice. This yields a precise revision of the GLS conjecture for representation-tame GLS algebras of affine type.