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

Abstract

Using Galois covering theory and equivariant techniques, we classify all connected representation-tame Geiß--Leclerc--Schröer (GLS) algebras in terms of their defining triples $(C,D,Ω)$. We then study the GLS algebras $H(\widetilde{CD}_n)$ with minimal symmetrizers, where $n\geq 2$ and $\widetilde{CD}_2=\widetilde{B}_2$. By realizing these algebras as basic algebras of $\mathbb{Z}_2$-skew group algebras of the string algebras $H(\widetilde{C}_{2n-2})$, 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 $τ$-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.