Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$: $$\{ e_T: T \text{ is a semi-standard Young tableau with $k-1$ rows and with entries in $[k]$} \}, $$ where $[k]=\{1,2,\ldots,k\}$, $e_T = \Delta_{T_1} \cdots \Delta_{T_m}$, $T_1, \ldots, T_m$ are columns of $T$, $\Delta_{T_i}$ is the minor of a $k \times k$ matrix with $1, \ldots, p$ rows, and $j_1, \ldots, j_p$ columns, $p$ is the number of rows of $T_i$ and $j_1,\ldots,j_p$ are entries of $T_i$. Are there some references about this? Thank you very much.

Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$: $$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and with entries in $[k]$} \}, $$ where $[k]=\{1,2,\ldots,k\}$, $e_T = \Delta_{T_1} \cdots \Delta_{T_m}$, $T_1, \ldots, T_m$ are columns of $T$, $\Delta_{T_i}$ is the minor of a $k \times k$ matrix with $1, \ldots, p$ rows, and $j_1, \ldots, j_p$ columns, $p$ is the number of rows of $T_i$ and $j_1,\ldots,j_p$ are entries of $T_i$. Are there some references about this? Thank you very much.