Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s\_1\dots s\_n$ with $s\_i$ simple reflections, then $s\_i(\gamma)=\gamma$ for $i=1\dots n$?


I can't see a reason why this shouldn't be true, but am unable to prove it.


Thanks!



Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$
This is definitely not true. For instance, already in $\Phi=B\_2$, each root has a root orthogonal to it, so for every root there is some nontrivial element (in fact, a reflection) of the Weyl group fixing it. But e.g. a simple root in $B\_2$ is not fixed by any simple reflection.


However, what you might want to know is the following. If we choose any point $v$ in the dominant chamber of our root system, then the stabilizer of $v$ is exactly the parabolic subgroup of $W$ generated by simple reflections that fix $v$. For a proof of this see Lemma 10.3B of Humphreys' "Introduction to Lie Algebras and Representation Theory" (<https://www.springer.com/us/book/9780387900537>).


