The Cluster Basis of ${\Bbb Z}[x_{1,1},\dots, x_{3,3}]$

Abstract

We show that the set of cluster monomials for the cluster algebra of type $D_4$ contains a basis of the ${\Bbb Z}$-module ${\Bbb Z}[x_{1,1},\dots,x_{3,3}]$. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables.