Let $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$.

Let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in $S_n$ are transpositions.

Let $W$ be a Coxeter group. What are all reflections in $W$? Thank you very much.

  • $\begingroup$ Are you interested in finite $W$ only? $\endgroup$ – Jules Lamers Nov 29 '16 at 15:39

As you find in "Humphreys, Reflection and Coxeter groups" (link behind paywall) in Section 5.7, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$, this is, all elements in $W$ that are conjugate to the generators $S$.

For the permutation group $\mathcal{S_n}$ of $\{1,\ldots,n\}$ generated by the set of simple transpositions $(i,i+1), 1 \leq i < n$, these then are exactly all transpositions $(i,j), 1 \leq i < j \leq n$.

A few more words: A priori, a Coxeter group $W$ is an abstract group together with a distinguished set $S$ of generators. In this sense, the above notion of reflections $R$ is well-defined but not related to any geometric notion of reflection you ask for. But given $(W,S)$, one can construct a geometric representation of $W$ on a real vector space with basis $\{ \alpha_s : s \in S\}$ (see Section 5.3 in the above reference). In this representation, one can now ask for those elements that have $|S|-1$ eigenvalues $1$ and one eigenvalue $-1$. And these are the exactly the reflections $R$ in the above sense.

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