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Let $V$ be a vector space over $\mathbb{R}$. An element $s \in GL(V)$ is a reflection if $H_s:=\ker(s-1)$ is a hyperplane and $s^2=1$. The eigenvalues of a reflection $s$ are $1, -1$. Every reflection $s$ can be written as $s(x) = x - \alpha_s(x)v_s$ for some linear form $\alpha_s \in V^*$ and $v_s \in V$. One has $\ker \alpha_s = H_s = \ker(s-1)$. The vector $v_s$ is an eigenvector of $s$ for the eigenvalue $-1$ if $\alpha_s(v_s)=2$.

Let $W$ be a finite Coxeter group acting as a reflection group on a Euclidean space $V$. The reflection length $\ell_R(w)$ of $w \in W$ is the minimal integer $m$ such that $w=r_1 \ldots r_m$, where $r_i$ are reflections in $W$.

The reflection length of $w \in W$ is $\ell_R(w) = \dim(\text{Im}(1-w))$.

Are there some references which describe the elements in a Coxeter group which have the longest reflection length? Thank you very much.

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    $\begingroup$ You should look up "absolute order." The maximal elements in absolute order are the Coxeter elements (=products of all the simple reflections in some order, and their conjugates). $\endgroup$
    – Sam Hopkins
    Commented Dec 19, 2018 at 17:19
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    $\begingroup$ Armstrong's oft-cited monograph has a good introduction to absolute order and Coxeter elements: arxiv.org/abs/math/0611106 $\endgroup$
    – Sam Hopkins
    Commented Dec 19, 2018 at 17:27
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    $\begingroup$ Sorry, what I said above was wrong: the Coxeter elements are not all the maximal elements wrt absolute order. Quoting from pg. 30 of Armstrong "In the case of the symmetric group, the Coxeter elements are precisely the maximal elements of the absolute order; in all other cases, the Coxeter elements are a proper subclass of maximal elements." $\endgroup$
    – Sam Hopkins
    Commented Dec 19, 2018 at 18:38

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