Wild Kronecker quivers in the remaining low-dimensional cases

Establish, for the n-Kronecker quiver with n=3 and with n=4, dimension vectors (p,q) for which the matrix semi-invariant algebra C[Mat_{p,q}^n]^G contains no polynomial or hypersurface separating set.

Background

The n-Kronecker quiver is wild for n≥3. Existing results in the paper show that C[Mat_{2,2}n]G has no hypersurface separating algebra for n≥5, while the cases n=3 and n=4 remain unresolved for the purpose of verifying the conjecture for this quiver.

The unresolved task is to find suitable dimension vectors (p,q) in each of these two cases that force every separating set to exceed the polynomial-or-hypersurface size threshold.

References

To verify the conjecture for this quiver it remains to show that, for $n=3$ and for $n=4$, there exists $p,q$ such that $C[_{p,q}n]G$ does not contain a polynomial or hypersurface separating set.

— The separating variety for matrix invariants  (2508.13865 - Elmer, 19 Aug 2025) in Example following Conjecture 1, Section 1.5, “Invariants and semi-invariants of quivers”