= Positive-root data for Cn
{c}
{title2=$\alpha_i=\varepsilon_i-\varepsilon_{i+1},\quad\alpha_n=2\varepsilon_n$}
Choose positive <roots of a root system> $\varepsilon_i-\varepsilon_j$, $\varepsilon_i+\varepsilon_j$ for $i<j$, and $2\varepsilon_i$. The <simple roots> are $\alpha_i=\varepsilon_i-\varepsilon_{i+1}$ for $i<n$ and $\alpha_n=2\varepsilon_n$. Their <coroots> are $\alpha_i^\vee=\varepsilon_i-\varepsilon_{i+1}$ and $\alpha_n^\vee=\varepsilon_n$. Therefore the <fundamental weights> and <Weyl vector> are
$$
\omega_i=\sum_{j=1}^i\varepsilon_j,\qquad \rho=\sum_{j=1}^n(n-j+1)\varepsilon_j.
$$
The <highest root> is $2\varepsilon_1=2\alpha_1+\cdots+2\alpha_{n-1}+\alpha_n$. These formulas include rank one.
Back to article page
