 A5 root system

The root system of the complex special linear Lie algebra ${\mathfrak{sl}}_6(\mathbb{C})$ is $\{e_i-e_j:1\le i\ne j\le 6\}$ in ${\sum }_i{x}_i=0$. Its simple roots $e_i-e_{i+1}$ give a five-node An Dynkin diagram. The defining fundamental representation has six weights $w_i=e_i-\frac{1}{6}{\sum }_j{e}_j$. The dual representation has weights $-w_i$, and the two-index antisymmetric representation has fifteen weights $w_i+w_j$, $i<j$, all with weight multiplicity one.