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Log-concavity of cluster algebras of type $A_n$

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arxiv 2408.03792 v2 pith:LSITQXW3 submitted 2024-08-07 math.RT math.COmath.GT

Zhichao Chen , Guanhua Huang , Zhe Sun This is my paper · ORCID
classification math.RTmath.COmath.GT
keywords basisclusterconstantslog-concavitystructurethetatypealgebra
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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.

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