On cluster variables of rank two acyclic cluster algebras

Kyungyong Lee

On cluster variables of rank two acyclic cluster algebras

Combinatorics 2010-08-13 v2

Authors: Kyungyong Lee

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 $\left(\begin{array}{cc} 0 & c -c & 0 \end{array}\right)$ , 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.

Keywords

cluster algebras cluster variables Laurent expression rank two acyclic quiver Grassmannians Euler-Poincaré characteristics

Cite

@article{arxiv.1008.1829,
  title  = {On cluster variables of rank two acyclic cluster algebras},
  author = {Kyungyong Lee},
  journal= {arXiv preprint arXiv:1008.1829},
  year   = {2010}
}

Comments

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)

On cluster variables of rank two acyclic cluster algebras

Abstract

Keywords

Cite

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