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A CONJECTURE ON $C$-MATRICES OF CLUSTER ALGEBRAS

Part of: Arithmetic rings and other special rings Algebraic combinatorics

Published online by Cambridge University Press:  19 June 2018

PEIGEN CAO ,
MIN HUANG  and
FANG LI
PEIGEN CAO
Affiliation:
School of Mathematical Sciences, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang 310027, PR China email peigencao@126.com
MIN HUANG
Affiliation:
Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada email minhuang1989@hotmail.com
FANG LI*
Affiliation:
School of Mathematical Sciences, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang 310027, PR China email fangli@zju.edu.cn
*
*Corresponding author.
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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.

MSC classification

Primary: 13F60: Cluster algebras 05E40: Combinatorial aspects of commutative algebra
Type
Article
Information
Nagoya Mathematical Journal , Volume 238 , June 2020 , pp. 37 - 46
Copyright
© 2018 Foundation Nagoya Mathematical Journal

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