As in the title, I am trying to label the roots in a root system $C_3$. One choice of simple roots is $\alpha_1 = (1,-1,0)$ , $\alpha_2 = (0,1,-1)$,$\alpha_3 = (0,0,2)$. The remaining positive root then are:
$\alpha_1 + \alpha_2 = (1,0,-1)$, $\alpha_2 + \alpha_3 = ( 0,1,1)$,
$\alpha_1 + \alpha_2 + \alpha_3 = (1,0,1)$, $2\alpha_2+\alpha_3=(0,2,0)$,
$\alpha_1+2\alpha_2+\alpha_3=(1,1,0)$, $2\alpha_1+2\alpha_2+\alpha_3=(2,0,0)$.
Now, we also now that the fundamentals reflection $s_i$ in the hyperplane orthogonal to $\alpha_i$ is given by:
\begin{equation}
s_i(\alpha_j) = \alpha_j – A_{ij}\alpha_i,
\end{equation}
where $A=(A_{ij})$ is a Corresponding Cartan matrix, in this case
\begin{pmatrix}
2 & -1 & 0\\
-1 & 2 & -1\\
0 & -2 & 2
\end{pmatrix}
We also know that the fundamental reflections $s_i$ satisfy $s_i(\Phi) = \Phi$ , where $\Phi$ is a corresponding root system.
Now, for example, if we do:
$s_2(2\alpha_2+\alpha_3)=2s_2(\alpha_2)+s_2(\alpha_3) = -2\alpha_2 + \alpha_3 + \alpha_2 = -\alpha_2 + \alpha_3$,
which is not in $\Phi$. Why is that? What am I doing wrong? Am I lebeling the roots incorrectly?
Best Answer
As you figure in your comment, either your source got the $i$ and $j$ flipped in the formula, or its Cartan matrix should be transposed. Either way, in the $C_3$ root system which your vectors seem to realise correctly in $\mathbb R^3$, we have $s_2(\alpha_3) = \alpha_3 +2\alpha_2$ (has to be a long root), and $s_3(\alpha_2)=\alpha_2+\alpha_3$. Note that one can even visualise this, e.g. the middle one here is $C_3$: https://en.m.wikipedia.org/wiki/Root_system#/media/File%3ARoot_vectors_b3_c3-d3.png