Let H(C,D,Omega) be a Geiß–Leclerc–Schröer algebra associated with a symmetrizable Cartan matrix C, symmetrizer D, and orientation Omega. Let Delta^+(C) denote the set of positive roots of the associated Kac–Moody root system. The GLS conjecture asserts that operatornamerank M| M is an indecomposable tau-locally free H(C,D,Omega)-module=Delta^+(C). The general assertion is false; the revised problem is to determine the rank vectors in the affine representation-tame cases, including the additional non-root vectors that must be added to Delta^+(C).

Geiß–Leclerc–Schröer conjecture

Let $H(C,D,\Omega)$ be a Geiß–Leclerc–Schröer algebra associated with a symmetrizable Cartan matrix $C$ , symmetrizer $D$ , and orientation $\Omega$ . Let $\Delta^+(C)$ denote the set of positive roots of the associated Kac–Moody root system. The GLS conjecture asserts that

\left\{\operatorname{rank} M\mid M\text{ is an indecomposable }\tau\text{-locally free }H(C,D,\Omega)\text{-module}\right\}=\Delta^+(C).

The general assertion is false; the revised problem is to determine the rank vectors in the affine representation-tame cases, including the additional non-root vectors that must be added to $\Delta^+(C)$ .

References

Primary source

arXiv

Additional references

Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems — arXiv — Qiang Dong, Zengqiang Lin, Ming Lu, Shiquan Ruan

lie-theory representation-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A 2026 preprint claims a corrected classification in a restricted affine setting, while the original general correspondence is known to fail.

The Geiß–Leclerc–Schröer conjecture proposes that the rank vectors of indecomposable $\tau$ -locally free modules coincide with the positive roots of the associated Kac–Moody algebra. The affine results show that this correspondence needs correction beyond the previously established special cases.

Known results

Lin and Su (2024): every positive affine root occurs as the rank vector of a $\tau$ -locally free module, but the converse fails in general.
Lin and Su (2024): explicit non-root rank vectors occur for affine types $B_n^{(1)}$ , $C_n^{(1)}$ , $F_4^{(1)}$ , and $G_2^{(1)}$ .
The correspondence is known for rigid $\tau$ -locally free modules and for type $\widetilde{C}_n$ with minimal symmetrizer.

September 22, 2026 classification

A preprint by Qiang Dong, Zengqiang Lin, Ming Lu, and Shiquan Ruan claims a complete classification for connected representation-tame GLS algebras. For type $\widetilde{CD}_n$ , it replaces the root-vector set by $\Delta^+\cup\{\beta_i+(2r+1)\delta\mid i=1,2,\ r\in\mathbb{Z}_{\ge 0}\}$ ; this claimed classification remains unverified.

Current status (as of September 2026): the original correspondence is disproved in several affine types, while the claimed corrected classification is unverified and the unrestricted GLS problem remains open.

Sources

No solutions have been posted yet.