Naturality of the rank-preserving bijection

Determine whether the rank-preserving bijection between square-zero upper triangular matrices and upper triangular matrices satisfying U^T U=0 can be made functorially compatible with every linear isomorphism and independent of a coordinate ordering.

Background

The paper’s explicit bijection is canonical only after fixing the standard coordinate order and using an RREF–lexicographic adapted-basis rule. The authors distinguish this flag-compatible combinatorial canonicity from stronger naturality under arbitrary linear isomorphisms.

They explicitly leave the stronger functorial and coordinate-independent version unresolved, while noting that the existing construction is natural in the weaker sense determined by the standard complete flag.

References

However, if ``natural'' is required to mean functorial compatibility with every linear isomorphism and complete independence of a coordinate ordering, then naturality in this stronger sense remains an interesting question.

— Singular Cholesky Fibers over Finite Fields  (2609.00533 - Wu et al., 1 Sep 2026) in Remark following Theorem 5.1, Section 5