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Chen–Huang–Sun log-concavity and Chen unimodality conjectures for cluster monomials

For every n≥1n\geq 1, every initial seed with cluster x=(x1,…,xn)\mathbf{x}=(x_1,\ldots,x_n) in a cluster algebra of type AnA_n, and every cluster monomial MM, write its Laurent expansion as M=∑a∈Zncax1a1⋯xnanM=\sum_{\mathbf{a}\in\mathbb{Z}^n}c_{\mathbf{a}}x_1^{a_1}\cdots x_n^{a_n}. The conjectures assert that the coefficient array (ca)(c_{\mathbf{a}}) is log-concave and unimodal, in the coefficient-array senses used by Chen–Huang–Sun and Chen, respectively. The assertions are required to hold independently of the choice of initial seed.

References

Primary source

arXiv

Additional references

algebraic-combinatoricscluster-algebras
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper proves both coefficient-pattern properties in special coordinate systems, but the general conjectures remain open.

The Chen–Huang–Sun conjectures ask whether Laurent-expansion coefficients of cluster monomials satisfy log-concavity and unimodality. The full assertions, including independence from the initial seed, remain open.

Known results

  • Chen–Huang–Sun established log-concavity for cluster variables of type AnA_n.
  • Chen–Huang–Sun established the cluster-monomial case for type A2A_2.
  • Chen and Sun, 2026: all cluster monomials of type A3A_3 are log-concave and unimodal, independently of the initial seed.
  • For type AnA_n with n≥4n\geq 4, the corresponding assertions and general seed independence remain conjectural.

October 2026 fan-coordinate advance

Chen and Sun report both properties for cluster monomials in fan initial-cluster coordinates of type AnA_n, using coefficient formulas and convolution arguments. This is a substantial coordinate-specific advance, not a solution of the full conjectures; the claim is unverified here.

Current status (as of October 2026): The type A3A_3 case and the reported fan-coordinate case are established in the cited papers, while the full higher-rank and seed-independent conjectures remain open.

Sources

Solutions 0

No solutions have been posted yet.