On cluster variables of rank two acyclic cluster algebras

Abstract

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 $\left(\begin{array}{ccc}0& c-c& 0\end{array}\right)$, and show that a large number of coefficients are non-negative. As a corollary, we obtain an explicit expression for the Euler-Poincaré characteristics of the corresponding quiver Grassmannians.

10 pages, thanks to referees, typos corrected, credits given, references added, follow-up is given in "Strong sign-coherency of certain symmetric polynomials, with application to cluster algebras" (arxiv preprint)

Cited by in corpus (3)

• A Combinatorial Formula for Rank 2 Cluster Variables
• Positivity in Quantum Cluster Algebras and Flags of Valued Quiver Representations
• Strong sign-coherency of certain symmetric polynomials, with application to cluster algebras