Questions about fundamental representations of $SL_3/U$. Consider the group $SL_3$. Let $U$ be the subgroup of $SL_3$ consisting of all upper triangular unipotent matrices. Then the algebra $\mathbb{C}[SL_3/U]$ is generated by $a_{11}, a_{21}, a_{31}, a_{11}a_{22}-a_{12}a_{21}, a_{11}a_{32}-a_{12}a_{31}, a_{21}a_{32}-a_{22}a_{31}$. There are two fundamental representations of $SL_3/U$: $V_{\omega_1}$ and $V_{\omega_2}$. Do we have the following result:  the heighest weight vector of $V_{\omega_1}$ is $a_{11}$ and the heighest weight vector of $V_{\omega_2}$ is $a_{11} a_{22}-a_{12}a_{21}$? Are there some references about representation theory of $SL_n$ and $SL_n/U$? Thank you very much.

Let $V_{\lambda} = \langle e_T : T \text{ is a semistandard Young tableau of shape $\lambda$} \rangle$, $\lambda$ is a Young diagram, $e_T = \prod e_c$, $c$ ranges over the columns of $T$, for $c = (c_1, \ldots, c_l)^T$, $e_c$ is the minor of $x=(x_{ij})$ consisting of the first $l$ columns and rows $c_1, \ldots, c_l$. Then $V_{\lambda}$ is an irreducible representation of $SL_n$.
Let $\lambda = (1,1,0)$ and $T$ a semistandard Young tableau of shape $\lambda$. Then $ T = \begin{matrix} 1 \\ 2 \end{matrix}$
and $e_T$ is the principle minor of $(x_{ij})$ consisting of the first two  rows and first two columns. Therefore fundamental representations are generated by principle minors.
