base affine space $G/N$ were described. A simple modification yields initial seeds for $\C[N]$ (see [@GLS3]). One of these seeds is $$((D_{1,2},\,D_{1,3},\,D_{12,23},\,D_{1,4},\,D_{12,34},\,D_{123,234}), Q),$$ where $Q$ is the triangular quiver: $$\xymatrix@-1.0pc{
&& \ar[dl]{1} && \\
&\ar[dl]{2}\ar[rr]&&{3}\ar[ul]\ar[dl]\\
{4}\ar[rr]&&{5}\ar[ul]\ar[rr]&&{6}\ar[ul]
}$$ Here, by $D_{I,J}$ we mean the regular function on $N$ assigning to a matrix its minor with row-set $I$ and column-set $J$. Moreover, the variables $$x_4=D_{1,4},\ x_5=D_{12,34},\ x_6=D_{123,234}$$ are *frozen*, they cannot be mutated, and therefore they belong to every cluster. If one performs the mutation $\mu_1$, one gets the new cluster variable $$\mu_1(x_1) = \frac{x_2+x_3}{x_1} = \frac{D_{1,3}+D_{12,23}}{D_{1,2}} = D_{2,3},$$ and thus the new seed $$((D_{2,3},\,D_{1,3},\,D_{12,23},\,D_{1,4},\,D_{12,34},\,D_{123,234}), \mu_1(Q)),$$ where $\mu_1(Q)$ is the mutated quiver: $$\xymatrix@-1.0pc{
&& \ar[dr]{1} && \\
&\ar[dl]{2}\ar[ur]&&{3}\ar[dl]\\
{4}\ar[rr]&&{5}\ar[ul]\ar[rr]&&{6}\ar[ul]
}$$ The full subquiver of $\mu_1(Q)$ obtained by erasing vertices $4,5,6$ corresponding to the frozen variables is a Dynkin quiver of type $A_3$, hence, by Theorem \[thFZ\], this cluster algebra has finitely many clusters and cluster variables. It is an easy exercise to check that indeed it has $14$ clusters and $12$ cluster variables if we count the $3$ frozen ones. Moreover, it follows from [@BZ0] that Lusztig’s dual canonical basis of $\C[N]$ coincides with the set of all cluster monomials.

Finally, note 