  Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular
matrices. In this paper, we show that the dual canonical basis of
$\mathbb{C}[N]$ (and base affine spaces) can be parameterized by semi-standard
Young tableaux. Moreover, we give an explicit formula for every element in the
the dual canonical basis using the data of the corresponding semistandard Young
tableau. We apply our results to study cluster variables in $\mathbb{C}[N]$.
