A CONJECTURE ON -MATRICES OF CLUSTER ALGEBRAS

Abstract

For a skew-symmetrizable cluster algebra ${\mathcal{A}}_{t_{0}}$ with principal coefficients at $t_{0}$, we prove that each seed $\unicode[STIX]{x1D6F4}_{t}$ of ${\mathcal{A}}_{t_{0}}$ is uniquely determined by its $C$-matrix, which was proposed by Fomin and Zelevinsky (Compos. Math. 143 (2007), 112–164) 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 Gross et al. (Canonical bases for cluster algebras, J. Amer. Math. Soc. 31(2) (2018), 497–608). Further discussion is provided in the sign-skew-symmetric case so as to obtain a weak version of the conjecture in this general case.