If I collect the Cartan generators of su(3) or sl(3,C), then the diagonal entries can be arranged as $$ \begin{pmatrix} 1 &1& -2 \\\ 0& 1 &-1 \end{pmatrix} $$
after a row operation this becomes $$ \begin{pmatrix} 1 & 0 & -1\\\ 0 & 1 & -1 \end{pmatrix} $$ which is the toric data of $CP^2$. I can then write the semigroup action/coordinate ring through the characters.
In fact, this generalizes to show the connection between $SU(n+1)$ and $CP^n$.
Now, how to relate this character to the character of repersentation of the group? this should be related to the highest weight rep.
