Signed cells for alternating principal-minor signs on symmetric matrices

Determine whether there exists a cluster structure on the variety of symmetric matrices for which the sign pattern (-1)^{|I|}det(M_I)<0 for every principal minor M_I defines a signed cell.

Background

The paper constructs a cluster structure on the coordinate ring of symmetric matrices and classifies its signed cells. It then compares these cells with the sign conditions (-1){|I|}det(M_I)\leq 0, which characterize, for nonnegative symmetric matrices, the property of having at most one positive eigenvalue.

The authors show that the constructed signed cells cannot realize this sign pattern: in particular, the cluster relations force the relevant 2-by-2 principal minor to be nonnegative. They therefore ask whether a different cluster structure on symmetric matrices could realize the desired strict sign pattern as a signed cell.

References

Is there another cluster structure on symmetric matrices where the sign pattern $(-1){|I|} \det M_I < 0$ gives a signed cell?

— Signed seeds and G-gradings on cluster algebras  (2610.03612 - Williams et al., 2 Oct 2026) in Question following the remark in Section 4, immediately before Section 5