Positivity in coefficient-free rank two cluster algebras

Details Positivity in coefficient-free rank two cluster algebras Positivity in coefficient-free rank two cluster algebras

2009-03-15

arxiv.org/abs/0903.2677v1

Mathematics

Rings and Algebras

9 pages

Scientific paper

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

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