A conjecture on $C$-matrices of cluster algebras

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 $\mathcal A_{t_0}$ with principal coefficients at $t_0$, we prove that each seed $\Sigma_t$ of $\mathcal A_{t_0}$ 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 $c$-vectors hold for $\mathcal A_{t_0}$, 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.