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

By the answer of the question, 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.

• Yes, as long as the root system is reduced. See Proposition 1.14 of Humphrey's "Reflection groups and Coxeter groups". – LSpice Dec 1 '16 at 14:03
• This "yes" only applies to finite Coxeter groups, I believe. – Christian Stump Dec 1 '16 at 14:35
• @ChristianStump, yes, you are right, although your answer shows that it's true in general. – LSpice Dec 7 '16 at 20:15

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.