Cluster Monomials as Simple Shifted Quantum Affine Modules

Definitions and mathematical statement

The supplied curation places this conjecture in the setting of simply-laced shifted quantum affine algebras. Let $\mathcal A_{\Gamma'_\infty}$ and $\mathcal O$ denote the cluster algebra and associated category intended in the source. A cluster monomial is a finite product \[ m=\prod_{i\in I}x_i^{a_i},\qquad a_i\in{\mathbb{Z}}_{\geq 0}, \] where the variables $x_i$ belong to a single cluster and only finitely many exponents are nonzero. The treatment of frozen variables and coefficients must follow the source's convention, which the supplied record does not specify. A simple object of $\mathcal O$ is a nonzero object with no nonzero proper subobject.

Write \[ \Phi:\mathcal A_{\Gamma'_\infty}\xrightarrow{\sim}R_{\mathcal O} \] for the isomorphism referred to in the source, oriented toward its category-side ring, and write $[L]$ for the class of an object $L$. Here $R_{\mathcal O}$ is only a placeholder for that intended target: the supplied record does not identify its precise construction.

Conjecture (source label [GHL1]). Within the source's intended setup, \[ \forall m\in\mathcal A_{\Gamma'_\infty}\text{ a cluster monomial},\quad \exists L\in\mathcal O\text{ simple such that }\Phi(m)=[L]. \] The supplied excerpt states that this conjecture is established for $\mathfrak g=\mathfrak{sl}_2$ in [GHL1]. This is a source-reported result, not an independently verified assertion here.

Unresolved setup. The record does not define $\Gamma'_\infty$, the shifted quantum affine algebras and their parameter restrictions, the precise category $\mathcal O$, or the isomorphism $\Phi$. Consequently, the displayed statement precisely records the claimed correspondence conditional on the intended source setup; it is not a self-contained specification of that setup.

Short English statement

The conjecture says that every cluster monomial, under the source's specified isomorphism, is the class of a simple object in the associated category $\mathcal O$. The supplied curation describes the general setting as simply-laced and reports the $\mathfrak{sl}_2$ case as proved. Essential details of the algebra, category, and isomorphism remain unspecified in this record.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006269$, source problem code catalogue entry, title "Cluster Monomials as Simple Shifted Quantum Affine Modules." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2025$; no individual proposer is identified.
• Enveloping Algebras and Geometric Representation Theory (2025), Oberwolfach Reports, pp. 1271--1332, DOI https://doi.org/10.4171/owr/2025/25. The supplied excerpt locates the conjecture on p. 1301, labels it [GHL1], and attributes the $\mathfrak g=\mathfrak{sl}_2$ result to [GHL1]. The report is described as documenting workshop discussions and open problems.
• Supplied literature source: "OWR 2025 report," https://ems.press/content/serial-article-files/51851.
• Supplied literature source: "Geiss--Hernandez--Leclerc, shifted quantum affine algebras," https://arxiv.org/abs/2401.04616. The supplied record does not reproduce the full bibliographic entry for [GHL1].
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved, reports the cluster-monomial correspondence as proved for $\mathfrak{sl}_2$, and describes the general simply-laced conjecture as unresolved. This is supplied dated curation derived from the OWR material, not independently verified here; its status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked, the question isolated from surrounding exposition, and notation repaired for self-contained publication. These are supplied provenance claims; no independent source visit or mathematical verification is asserted here.