Positivity for cluster algebras of rank 3 (Q374588)

Cluster algebras, invented by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups, are a class of commutative algebras endowed with a distinguished set of generators, the cluster variables. The cluster variables are grouped into finite subsets, called clusters, and are defined recursively from initial variables through mutation on the clusters. Finding explicit computable direct formulas for the cluster variables is one of the main open problems in the theory of cluster algebras and has been studied by many mathematicians. Fomin and Zelevinsky [loc. cit.] showed that every cluster variable is a Laurent polynomial in the initial variables \(x_1,x_2,\dots, x_n\), and they conjectured that this Laurent polynomial has positive coefficients.NEWLINENEWLINEIn this paper, the authors prove that the positivity conjecture holds in every skew-symmetric coefficient-free cluster algebra of rank 3. It is great to know that the authors prove that the positivity conjecture holds in every skew-symmetric cluster algebra in the paper titled ``Positivity for cluster algebras'' which will appear in Annals of Mathematics.