Questions about fundamental representations of $SL_3/U$.

Question

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.

Answer 1

Given a complex semisimple group $G$ with Borel $B=TU$, where $T$ is a maximal torus and $U$ is the unipotent radical of $B$, the representation $\mathbf{C}[G/U]$ is a multiplicity-free direct sum over the irreducible finite dimensional representations of $G$. This is a consequence of the following algebraic version of the Peter-Weyl theorem:

$$\mathbf{C}[G] \cong \bigoplus V \otimes V^*,$$ in which the isomorphism is of $G$-bimodules and the sum runs over all finite dimensional irreducible $G$-modules $V$; in turn the algebraic Peter-Weyl theorem is a consequence of the fact that each $G$-module is semisimple and that $G$ acts locally finitely on $\mathbf{C}[G]$. Each $V^*$ is a highest weight (right) module, which means its space of $U$-invariants is one dimensional, implying the claim that $\mathbf{C}[G/U]$ is a multiplicity-free sum of all the finite dimensional irreducible $G$-modules.

For the particular case in your question, one needs precise definitions of the functions $a_{ij}$; I believe that your assertions are correct with a certain convention for the indexing.

For more information, you should search the internet using the keyword basic affine space to refer to $G/U$.

Answer 2

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.