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Positivity for cluster algebras. (English) Zbl 1350.13024
Cluster algebras, invented by S. Fomin and 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 \(x_1, x_2,\dots, x_N\) through mutation on the clusters, which is determined by the choice of a skew-symmetric \(N\times N\) integer matrix \(B\), or, equivalently, by a quiver \(Q\). The seed includes the cluster variables and the iterative process or mutation is codified in an exchange matrix. Cluster algebras are defined via matrices and mutations and also exchange patterns via matrices and polynomials. Fomin and Zelevinsky 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 [loc. cit.]. In the paper under review, the authors prove the positivity conjecture for all skew-symmetric cluster algebras. It is worth mentioning that the authors prove positivity almost exclusively by elementary algebraic computations, and the advantage of this approach is that there is no need to restrict to a special type of cluster algebras.

MSC:
13F60 Cluster algebras
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