A conjecture on $C$-matrices of cluster algebras
Peigen Cao, Min Huang, Fang Li

TL;DR
This paper proves that in skew-symmetrizable cluster algebras with principal coefficients, each seed is uniquely determined by its C-matrix, confirming a conjecture by Fomin and Zelevinsky.
Contribution
It establishes the uniqueness of seeds via C-matrices in skew-symmetrizable cluster algebras, extending the understanding of their structure.
Findings
Each seed is uniquely determined by its C-matrix.
Positivity of cluster variables holds in the studied case.
Sign-coherence of c-vectors is verified for the algebra.
Abstract
For a skew-symmetrizable cluster algebra with principal coefficients at , we prove that each seed of is uniquely determined by its {\bf C-matrix}, which was proposed by Fomin and Zelevinsky in \cite{FZ3} as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign-coherence of -vectors hold for , which was actually verified in \cite{GHKK}. More discussion is given in the sign-skew-symmetric case so as to obtain a conclusion as weak version of the conjecture in this general case.
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