Dual canonical bases for unipotent groups and base affine spaces

Abstract

Denote by [Formula: see text] the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of [Formula: see text] (and base affine spaces) can be parametrized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the dual canonical basis using the data of the corresponding semi-standard Young tableau. We apply our results to study cluster variables in [Formula: see text].