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{\bf Mark Skandera}
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{\bf The Cluster Basis of ${\Bbb Z}[x_{1,1},\dots, x_{3,3}]$}
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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.
\bye