= Fundamental weights of sl3
{title2=$\omega_1=\varepsilon_1,\quad\omega_2=-\varepsilon_3$}
For the <special linear Lie algebra> $\mathfrak{sl}_3(\mathbb C)$ with upper triangular <positive roots>, the <simple coroots> are $\operatorname{diag}(1,-1,0)$ and $\operatorname{diag}(0,1,-1)$. Their dual <fundamental weights> are $\omega_1(\operatorname{diag}(t_1,t_2,t_3))=t_1$ and $\omega_2(\operatorname{diag}(t_1,t_2,t_3))=t_1+t_2=-t_3$. In simple-root coordinates, $\omega_1=(2\alpha_1+\alpha_2)/3$ and $\omega_2=(\alpha_1+2\alpha_2)/3$. The primitive vectors $e_1\in\mathbb C^3$ and $e_1\wedge e_2\in\Lambda^2\mathbb C^3$ realize these <Fundamental representations of sl3>.
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