By the answer of [the question](https://mathoverflow.net/questions/255919/references-request-reflections-in-coxeter-groups), the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$.


Do we have a one to one correspondence between positive roots and reflections in a Coxeter group?


Thank you very much.



Do we have a one to one correspondence between positive roots and reflections in a Coxeter group?
I suggest, you really look carefully into the reference "Humphreys, Reflection and Coxeter groups" ([link behind paywall](http://www.ams.org/mathscinet-getitem?mr=1066460)), as you again find the answer there.
Please look into the reference for details below. In case, you have no access to it, you might be lucky searching for it on the web (you can also drop me a line).


First, one has to define a root system for $W$ (done in Section 5.4) as $\Phi = \{w(\alpha\_s) : w \in W, s \in S\}$. Then there is a decomposition $\Phi = \Phi^+ \sqcup -\Phi^+$ with $\Phi^+ = \Phi \cap \mathbb{R}\_{\geq 0}\Delta$ for $\Delta = \{\alpha\_s : s \in S\}$.
And finally, by construction, $R = \{ s\_\beta : \beta \in \Phi^+\}$ and this is a bijective correspondence. So the answer to your question is **yes**.


