Dupont, GrĂ©goire Positivity in coefficient-free rank two cluster algebras. (English) Zbl 1193.16017 Electron. J. Comb. 16, No. 1, Research Paper R98, 11 p. (2009). 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. Cited in 3 Documents MSC: 16G20 Representations of quivers and partially ordered sets 13F60 Cluster algebras 05E10 Combinatorial aspects of representation theory Keywords:rational functions; substraction-free Laurent polynomials; Fomin-Zelevinsky positivity conjecture; rank two cluster algebras PDFBibTeX XMLCite \textit{G. Dupont}, Electron. J. Comb. 16, No. 1, Research Paper R98, 11 p. (2009; Zbl 1193.16017) Full Text: arXiv EuDML EMIS