Log-concavity and unimodality of cluster monomials of type $A_3$

TL;DR

This paper proves the log-concavity and unimodality of cluster monomials of type A3, extending known results from lower ranks and exploring related conjectures in cluster algebra theory.

Contribution

It establishes the log-concavity and unimodality for type A3 cluster monomials, a complex case previously unproven, and refines related conjectures.

Findings

Proved log-concavity of A3 cluster monomials.

Established unimodality of A3 cluster monomials.

Extended conjectures on unimodality and isomorphism in cluster algebras.

Abstract

The log-concavity of cluster variables of type $A_n$ and cluster monomials of type $A_2$ was established by Chen-Huang-Sun. It is still a conjecture for the cluster monomials of higher rank. In this paper, we prove the log-concavity and unimodality of the cluster monomials of type $A_3$, a substantially more intricate case. Moreover, we refine and extend this conjecture by considering the unimodality and the strongly isomorphism of cluster algebras.