Dual canonical basis for unipotent group and base affine space

Jian-rong Li

Published 2020-10-14Version 1

Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\mathbb{C}[N]$ can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis. Let $N^- \subset SL_k$ be the subgroup of unipotent lower-triangular matrices and let $\mathbb{C}[SL_k]^{N^-}$ be the coordinate ring of the base affine space $SL_k/N^{-}$. Denote by $\widetilde{\mathbb{C}[SL_k]^{N^-}}$ the quotient of $\mathbb{C}[SL_k]^{N^-}$ by identifying the leading principal minors with $1$. We also give an explicit description of the dual canonical basis of $\widetilde{\mathbb{C}[SL_k]^{N^-}}$ and give a conjectural description of the dual canonical basis of $\mathbb{C}[SL_k]^{N^-}$.