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

@misc{dong2026representationtame,
  title = {{Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems}},
  author = {Qiang Dong and Zengqiang Lin and Ming Lu and Shiquan Ruan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26503},
  eprint = {2609.26503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26503},
}

TY  - UNPB
TI  - Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems
AU  - Qiang Dong
AU  - Zengqiang Lin
AU  - Ming Lu
AU  - Shiquan Ruan
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26503
UR  - https://arxiv.org/abs/2609.26503
AB  - 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(CD̃_n) with minimal symmetrizers, where n≥ 2 and CD̃_2=B̃_2. By realizing these algebras as basic algebras of ℤ_2-skew group algebras of the string algebras H(C̃_2n−2), we introduce extended strings and extended bands to parameterize the connected components of the Auslander--Reiten quivers of H(CD̃_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.
ER  -