Log-concavity of cluster algebras of type $A_n$

TL;DR

This paper proves the log-concavity of coefficients of cluster variables and structure constants in type A_n cluster algebras, supporting a broader conjecture about their general log-concavity.

Contribution

It establishes log-concavity for coefficients of cluster variables in type A_n and for structure constants in type A_2, and conjectures this property for larger types.

Findings

Coefficients of cluster variables in type A_n are log-concave.

Structure constants for theta basis in type A_2 are log-concave.

Conjecture that log-concavity holds for all types in the theta basis.

Abstract

Okounkov [Oko03] conjectured the log-concavity about the structure constants for many interesting basis from representation theory. For the cluster algebra, Gross, Hacking, Keel, Kontsevich [GHKK18] introduced the atomic theta basis. We prove that the coefficients of the exponents of any cluster variable of type $A_n$ are log-concave. We show that the structure constants for theta basis of type $A_2$ are log-concave. As for larger generality, we conjecture that the log-concavity of the structure constants for theta basis of the cluster algebra.