A CONJECTURE ON $C$-MATRICES OF CLUSTER ALGEBRAS. (19th June 2020)

• Record Type: Journal Article
• Authors: CAO, PEIGEN; HUANG, MIN; LI, FANG
• Is Part Of: Nagoya mathematical journal. Volume 238(2020)
• Page Start: 37
• Page End: 46
• Publication Date: 2020-06-19
• Subjects: 13F60, -- 05E40; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathematics; Wiskunde; Electronic journals; Periodicals; 510.5
• DOI: 10.1017/nmj.2018.18
• Languages: English
• ISSNs: 0027-7630

Abstract

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.