Quiver Discriminants from Generic Maximal Minors

Definitions and mathematical statement

Let $n\geq3$ and let $X=(x_{ij})$ be a generic $(n-2)\times(n-1)$ matrix. Denote by $M_j$ the determinant obtained by deleting column $j$. The source calls $f$ a free divisor "formed by the maximal minors" but does not specify the operation. The natural product interpretation is \[ f(X)=\prod_{j=1}^{n-1}M_j(X), \] up to a nonzero scalar and reduction of the equation if needed; this interpretation must be checked against the report. The ideal generated by all $M_j$ defines a determinantal locus, not the same object as their product divisor, so the two cannot be interchanged.

In the usual characteristic-zero algebraic setting, a reduced hypersurface $f=0$ in an $N$-dimensional affine space is a free divisor if its logarithmic derivation module \[ \operatorname{Der}(-\log f)=\{\delta\in\operatorname{Der}_k(k[x_1,\ldots,x_N]): \delta(f)\in(f)\} \] is locally free of rank $N$. The source does not specify the coefficient field or analytic versus algebraic convention.

For a finite quiver $Q=(Q_0,Q_1)$ and a dimension vector $\mathbf d=(d_i)_{i\in Q_0}$, the representation variety and base-change group are \[ \operatorname{Rep}(Q,\mathbf d)= \bigoplus_{a\in Q_1}\operatorname{Hom}(k^{d_{s(a)}},k^{d_{t(a)}}), \qquad G_{\mathbf d}=\prod_{i\in Q_0}\operatorname{GL}_{d_i}(k), \] with $(g\cdot A)_a=g_{t(a)}A_ag_{s(a)}^{-1}$. When there is a dense open orbit, its complement is the discriminant; for the intended realization this complement must be a reduced hypersurface identified with $f=0$.

The classification request is to determine all relevant $(Q,\mathbf d)$ for which that discriminant is the specified maximal-minor free divisor. To make it precise one must state whether the identification is by a linear coordinate isomorphism, arbitrary algebraic isomorphism, analytic equivalence, or a slice construction; one must also state restrictions on quivers and conventions about zero-dimensional or extraneous vertices. These choices are absent and are not silently selected.

The supplied literature triage checked 21 August 2026 says the report presents a $D_4$ example as one realization and asks for possible quivers and dimension vectors more generally. It gives no orientation or dimension vector for that example and reports no complete classification in the checked literature. Neither those missing data nor a general classification are supplied here.

Short English statement

Classify the quivers and dimension vectors whose open-orbit discriminants realize the stated maximal-minor divisor. The product of maximal minors is a natural but unconfirmed reading of the missing formula, and the equivalence notion is unspecified. The supplied assessment records a $D_4$ example without a complete classification.

Sources

• Local UnsolvedMath registry, permanent ID 30000181, exact source code catalogue entry; Quiver Discriminants from Generic Maximal Minors. Proposed year 2005, status open; corrected_verified extraction retains the free-divisor classification question.
• Representation Theory of Finite-Dimensional Algebras (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/06, pp. 319--406, https://doi.org/10.4171/owr/2005/06.
• Supplied literature triage checked 2026-08-21, citing only "original OWR report", https://doi.org/10.4171/owr/2005/06. It reports one $D_4$ realization and no complete classification for the stated exercise. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no separate later-paper citation is supplied.