A conjecture on $C$-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 ${\Sigma }_t$ 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.

7 pages

Cited by in corpus (1)

• Cluster realizations of Weyl groups and higher Teichmüller theory