AI 中文总结
本文分类了表示驯服的GLS代数，研究其最小对称化子情形，通过扩展弦带描述AR箭图，并修正了仿射型GLS猜想，证明秩向量对应正根及非根向量。
AI 中文摘要
利用Galois覆盖理论与等变技巧，我们根据定义三元组$(C,D,\Omega)$对所有连通的表示驯服Geiß--Leclerc--Schröer (GLS)代数进行了分类。随后，我们研究了具有最小对称化子的GLS代数$H(\widetilde{CD}_n)$，其中$n\geq 2$且$\widetilde{CD}_2=\widetilde{B}_2$。通过将这些代数实现为弦代数$H(\widetilde{C}_{2n-2})$的$\mathbb{Z}_2$-斜群代数的基本代数，我们引入了扩展弦和扩展带，用以参数化$H(\widetilde{CD}_n)$的Auslander--Reiten箭图的连通分支并确定其形状。我们进一步对仿射型表示驯服GLS代数的不可分解$\tau$-局部自由模进行了分类，并证明其秩向量恰好构成正根集合，连同正锥中明确描述的非根向量。这给出了仿射型表示驯服GLS代数的GLS猜想的一个精确修正。
英文摘要
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.
Comments47 pages; comments are welcome
