# Find the highest weight and highest weight vector for adjoint representation of $\mathfrak{sl}(n)$

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?

Take $$E_{1n}$$. Then, whenever $$i, $$[E_{ij},E_{1n}]=0$$.
• 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?