Positive-root data for Cn (${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1},{\alpha }_n=2{\epsilon }_n$)

Choose positive roots of a root system ${\epsilon }_i-{\epsilon }_j$, ${\epsilon }_i+{\epsilon }_j$ for $i<j$, and $2{\epsilon }_i$. The simple roots are ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n=2{\epsilon }_n$. Their coroots are ${\alpha }_i^{\vee }={\epsilon }_i-{\epsilon }_{i+1}$ and ${\alpha }_n^{\vee }={\epsilon }_n$. Therefore the fundamental weights and Weyl vector areThe highest root is $2{\epsilon }_1=2{\alpha }_1+\cdots +2{\alpha }_{n-1}+{\alpha }_n$. These formulas include rank one.