= C3 root system
{c}
{title2=$C_3$}
The $C_3$ roots are $\pm\varepsilon_i\pm\varepsilon_j$ for $1\leq i<j\leq3$ and $\pm2\varepsilon_i$ for $1\leq i\leq3$. One root basis is $\alpha_1=\varepsilon_1-\varepsilon_2$, $\alpha_2=\varepsilon_2-\varepsilon_3$, $\alpha_3=2\varepsilon_3$. With $A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle$, its <Cartan matrix> is
$$
\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{pmatrix}.
$$
The fundamental representation of highest weight $[1,0,0]$ is six-dimensional, with weights $\pm\varepsilon_1,\pm\varepsilon_2,\pm\varepsilon_3$.
Back to article page
