By definition, the highest weight vector for $\mathfrak{sl}(n)$ is an eigenvector for the action of a Cartan subalgebra which is killed by the adjoint action of $E_{ij}$ with $i < j$ (which correspond to the eigenspaces of positive roots). Let us choose, as a Cartan subalgebra, the subalgebra of all diagonal matrices of $\mathfrak{sl}(n)$. Now, the highest weight vector must be of type $E_{kh}$ with $k \neq h$ because all eigenvectors of non zero eigenvalue are elementary matrices of this type. What I find most difficult now is to find an elementary matrix $E_{kh}$ such that $[E_{ij}, E_{kh}] = 0$ for every $i < j$, because this happens everytime $j \neq k$ and $h \neq i$. What am I missing here? Is this even the right way to go about this problem?
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Take $E_{1n}$. Then, whenever $i<j$, $[E_{ij},E_{1n}]=0$.
answered Sep 2, 2021 at 17:54
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1$\begingroup$ Right, thank you. So, if $\epsilon_i(h)= h_i$ where $h_i$ is the $i$-th entry of a diagonal matrix $h$, the highest weight is $\epsilon_1 - \epsilon_n$, is that right? $\endgroup$– cipCommented Sep 2, 2021 at 18:09
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