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Subtraction-free characterization conjecture for bounded cluster-variable ratios

About 2 years old · traced to

Let D\mathcal{D} be a full-rank cluster algebra of finite type, and consider ratios of products of its cluster variables on the totally positive locus. Such a ratio is called bounded when it is bounded there. A ratio p/qp/q is subtraction free in cluster variables if q−pq-p can be expressed as a polynomial with positive coefficients in the full set of cluster variables. Cluster-variable subtraction-free conjecture. Every bounded ratio is subtraction free in cluster variables. The paper presents this as the natural cluster-algebra analogue of the determinantal-ratio conjecture, but supplies an explicit counterexample showing that it is false in general.

References

Primary source

Michael Gekhtman, Zachary Greenberg and Daniel Soskin, “Multiplicative Inequalities In Cluster Algebras Of Finite Type”, arXiv:2409.06642 (2024).

cluster-algebrastotal-positivity
Posted

Progress summary

Refreshed
Claimed solved

A 2024 paper gives an explicit counterexample, so the conjecture is claimed false rather than open, although the claim has not been independently verified here.

The conjecture asserts that every bounded ratio of products of cluster variables in a full-rank finite-type cluster algebra has a positive polynomial expression in all cluster variables. The paper Multiplicative Inequalities in Cluster Algebras of Finite Type states that this is false and supplies an explicit counterexample.

Known results

  • Every bounded ratio in a full-rank finite-type cluster algebra is bounded by 11.
  • Every extreme ray of the bounded cone is subtraction free in the full set of cluster variables.
  • The counterexamples arise from bounded integer ratios not represented by integer powers of extreme rays.

September 10, 2024 counterexample

The paper gives, in type C2C_2, the bounded ratio v2v4v6=x1x3x5x2x4x6\sqrt{v_2v_4v_6}=\frac{x_1x_3x_5}{x_2x_4x_6}. It is an integer combination of cluster variables but is not subtraction free: in the initial cluster, q−pq-p has a negative Laurent coefficient. This claims a complete disproof of the conjecture, but the claim is unverified here.

Current status (as of September 2026): The conjecture is claimed false by the explicit type C2C_2 counterexample; no later source reports a correction, but the disproof remains unverified in this report.

Sources

Solutions 0

No solutions have been posted yet.