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Positivity in coefficient-free rank two cluster algebras. (English) Zbl 1193.16017

Summary: Let \(b,c\) be positive integers, \(x_1,x_2\) be indeterminates over \(\mathbb{Z}\) and \(x_m\), \(m\in\mathbb{Z}\) be rational functions defined by \(x_{m-1}x_{m+1}=x^b_m+1\) if \(m\) is odd and \(x_{m-1}x_{m+1}=x^c_m+1\) if \(m\) is even. In this short note, we prove that for any \(m,k\in\mathbb{Z}\), \(x_k\) can be expressed as a substraction-free Laurent polynomial in \(\mathbb{Z}[x^{\pm 1}_m,x^{\pm 1}_m]\). This proves Fomin-Zelevinsky’s positivity conjecture for coefficient-free rank two cluster algebras.

MSC:

16G20 Representations of quivers and partially ordered sets
13F60 Cluster algebras
05E10 Combinatorial aspects of representation theory
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