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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?

amator2357
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    Strange...here https://math.stackexchange.com/questions/2947216/a-question-about-weyl-group-action-in-the-book-of-bourbaki?rq=1 the definition of $s_i(\alpha_j)$ is $s_i(\alpha_j)=\alpha_j - A_{ji}\alpha_i$..is my definition of reflection wrong? In all the sources that I have seen it is given in the form that I gave... – amator2357 Jun 27 '19 at 22:57

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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

Torsten Schoeneberg
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