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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.

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  • $\begingroup$ Yes, as long as the root system is reduced. See Proposition 1.14 of Humphrey's "Reflection groups and Coxeter groups". $\endgroup$
    – LSpice
    Commented Dec 1, 2016 at 14:03
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    $\begingroup$ This "yes" only applies to finite Coxeter groups, I believe. $\endgroup$
    – Christian Stump
    Commented Dec 1, 2016 at 14:35
  • $\begingroup$ @ChristianStump, yes, you are right, although your answer shows that it's true in general. $\endgroup$
    – LSpice
    Commented Dec 7, 2016 at 20:15

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I suggest, you really look carefully into the reference "Humphreys, Reflection and Coxeter groups" (link behind paywall), 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.

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    $\begingroup$ Note that the definition of "reflection" here is pretty far from any specific kind of geometry, but within the abstract theory of Coxeter groups it works surprisingly well as a generalization of the usual notion for finite real reflection groups acting on a euclidean space. $\endgroup$
    – Jim Humphreys
    Commented Dec 1, 2016 at 18:38

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