Positivity of Quiver Laurent Polynomials $\Omega_\alpha$

Definitions and mathematical statement

A quiver is a directed graph $Q=(Q_0,Q_1,s,h)$, where $Q_0$ is its vertex set, $Q_1$ is its arrow set, and $s,h:Q_1\to Q_0$ give the source and target of each arrow. An element $\alpha\in\mathbb N^{Q_0}$ assigns a natural number to each vertex. The supplied record does not specify finiteness restrictions on $Q$, restrictions on loops or multiple arrows, or whether $\mathbb N$ includes zero; these conventions remain unresolved. No exclusion of the zero dimension vector is stated.

Let $t=q^{1/2}$ be a formal indeterminate. A Laurent polynomial in $t$ with nonnegative integer coefficients is a finite sum \[ \sum_{k\in\mathbb Z}c_k t^k,\qquad c_k\in\mathbb Z_{\geq 0}, \] where only finitely many coefficients are nonzero.

The conjectural target reported in the source record is \[ \text{for every quiver }Q\text{ and every }\alpha\in\mathbb N^{Q_0},\qquad \Omega_\alpha(-t)\in\mathbb Z_{\geq 0}[t,t^{-1}]. \] Here $\Omega_\alpha$ is the source's notation, whose dependence on $Q$ is implicit. The supplied record gives no definition or normalization of $\Omega_\alpha$, no ambient algebraic object in which it is initially defined, and no construction explaining the substitution $-t$. Consequently this display preserves the reported conjecture but is not a self-contained mathematical statement. A precise formulation requires the source definition of $\Omega_\alpha$ and its applicable quiver and dimension-vector conventions; none is inferred here.

Short English statement

The reported conjecture asks whether, for every quiver and every permitted dimension vector, substituting $-q^{1/2}$ into the source invariant $\Omega_\alpha$ produces a Laurent polynomial in $q^{1/2}$ with nonnegative integer coefficients. The invariant and its domain conventions are missing, so the formulation remains incomplete and requires human mathematical review.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30001718$, source problem code catalogue entry, titled "Positivity of Quiver Laurent Polynomials $\Omega_\alpha$." The record belongs to the Oberwolfach Reports Open Problems collection and assigns the proposed year 2011; no individual proposer is supplied.
• Supplied report citation: Representation Theory of Quivers and Finite Dimensional Algebras (2011), Oberwolfach Reports, pp. 523--608, DOI 10.4171/owr/2011/10. Source URL and supplied literature-source link, labeled "Oberwolfach Report 10/2011": https://doi.org/10.4171/owr/2011/10. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise location for this conjecture within the report is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the extraction contains only the heading and opening sentence of a combinatorial-positivity conjecture, without a complete mathematical claim, so it could not be verified as solved or disproved. This is supplied dated curation, not an independently verified assessment. The record expressly calls for independent verification of time-sensitive status and cited claims.
• The registry also labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the cleanup is self-contained and adds no unsupported source-dependent content. This supplied verification claim does not resolve the missing definition of $\Omega_\alpha$.