On cluster variables of rank two acyclic cluster algebras

Details On cluster variables of rank two acyclic cluster algebras On cluster variables of rank two acyclic cluster algebras

2010-08-11

arxiv.org/abs/1008.1829v2

Mathematics

Combinatorics

10 pages, thanks to referees, typos corrected, credits given, references added, follow-up is given in "Strong sign-coherency o

Scientific paper

In this note, we find an explicit formula for the Laurent expression of cluster variables of coefficient-free rank two cluster algebras associated with the matrix $\left(\begin{array}{cc} 0 & c -c & 0 \end{array}\right)$, and show that a large number of coefficients are non-negative. As a corollary, we obtain an explicit expression for the Euler-Poincar\'{e} characteristics of the corresponding quiver Grassmannians.

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