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For any finite crystallographic reflection group $W = \langle s_1, \ldots , s_n\rangle$, every hyperplane reflection is of the form $ws_iw^{-1}$ for some $i$ and some $w \in W$.

A finite crystallographic reflection group $W$ is a Coxeter group with the presentation \begin{align}\label{Coxeter system} S=\langle s_1, s_2, \ldots, s_n \mid (s_is_j)^{m_{ij}} = 1 \rangle, \end{align} where $(s_i)_{1 \leq i \leq n}$ is the set of simple reflections and $m_{ij} \in \{2,3,4,6\}$. The pair $(W,S)$ is called a Coxeter system.

I have some questions:

  1. Is it true that every finite reflection group consists of some (not necessarily hyperplane) reflections and some rotations?
  2. What are the reduced words of reflections under a Coxeter system $(W,S)$?

For any finite reflection group, the number of hyperplane reflections is the number of positive roots in the corresponding root system, see section 1.14 of Jim Humphreys' book "Reflection groups and Coxeter groups".

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    $\begingroup$ Not clear to me what "rotation" would mean, abstractly, ... $\endgroup$
    – paul garrett
    Commented Dec 20, 2016 at 18:58
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    $\begingroup$ Well, anyways, every reflection has some reduced word of that form (palindromic). This paper studies reduced words for reflections in Coxeter groups in detail: deepblue.lib.umich.edu/handle/2027.42/46149 $\endgroup$
    – Sam Hopkins
    Commented Dec 20, 2016 at 20:35
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    $\begingroup$ This wikipedia page has a terrible definition of "rotation." I was under the impression that wikipedia was starting to be somewhat reliable for math...this is the first math page I've seen in a long time that I think should be completely scrapped. A "Euclidean" rotation should be defined as an rigid motion whose fixed space is a codimension-2 subspace $\endgroup$
    – Nathan Reading
    Commented Dec 21, 2016 at 15:08
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    $\begingroup$ As @NathanReading says, that wiki page is terrible, and misleading. I am suspecting that the question's intent involves the standard representation of a Coxeter group, but perhaps one should say so? And, still, a "rotation" should have a codimension-two fixed subspace. $\endgroup$
    – paul garrett
    Commented Dec 21, 2016 at 22:21
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    $\begingroup$ @bing: Concerning your final sentence, the crystallographic restriction isn't needed for a finite Coxeter group (= finite group generated by reflections acting on real euclidean space) if you define "root" appropriately in that setting. Reflections are always relative to roots in this general setting, e.g., see the exposition in section 1.14 of my 1990 book. $\endgroup$
    – Jim Humphreys
    Commented Mar 16, 2017 at 17:59

2 Answers 2

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1) No. $W$ consists of elements of determinant 1 and -1. According to your wikpedia, all elements of determinant 1 are "rotations". Elements of determinant -1 are not necessarily reflections because they are not necessarily of order 2. Just think of a 4-cycle $(1,2,3,4)\in S_4$: his order is 4, not 2. It is a proper roto-reflection...

BTW, the terminology is confusing: a rotation can be a reflection! Think of a reflection across 2-codimensional subspace.

2) It is explained by Sam for (hyperplane) reflections. They are all of the form $ws_iw^{-1}$ for some $w\in W$, and they will have a reduced word of this kind.

Higher-dimensional (fixing a subspace of higher codimension) reflections can be figured out as well. They are just elements of order 2. I do not know their reduced words off the top of my head.

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  • $\begingroup$ A specific example (in $\mathbb{R}^3$) of an element of determinant $-1$ that's not a reflection is the composition of a reflection in the $xy$ plane with a rotation of $\frac{2\pi}n$ about the $z$ axis - think of it as a sort of rotary 'glide-reflection'; it'll have order $n$ if $n$ is even, or order $2n$ if $n$ is odd. $\endgroup$
    – Steven Stadnicki
    Commented Dec 21, 2016 at 16:35
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    $\begingroup$ In the theory of abstract Coxeter groups, "reflections" are usually defined to be the elements in the set $\{wsw^{-1}\colon w \in W, s\in S\}$ (where $(W,S)$ is some Coxeter system). Thus, reflections across codimension $>1$ subspaces are not usually called reflections in this context. $\endgroup$
    – Sam Hopkins
    Commented Dec 23, 2016 at 18:20
  • $\begingroup$ And by the way, I think it requires at least a small argument to show that every reflection has a reduced word that is a palindrome. $\endgroup$
    – Sam Hopkins
    Commented Dec 23, 2016 at 18:21
  • $\begingroup$ This homework solution sheet seems to provide such an argument: math.sfsu.edu/federico/Clase/Coxeter/HomeworkSolutions/9.pdf $\endgroup$
    – Sam Hopkins
    Commented Dec 23, 2016 at 18:25
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Every coxeter group consists of reflections. An even number of reflections around two elements is a rotation, which is what $(s_is_j)^{m_{ij}}$ means. The infinite coxeter groups consists of sets of parallel mirrors, which equate to a translation rather than a reflection.

One might note the root lattice consists of perpendiculars to each of the mirrors, radiating from a point. This produces a lattice.

The 'reduced word' is the shortest path between two points. So if $ABABAB=1$ as in the hexagon, then $ABAB=BA$, where $BA$ is the reduced form.

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  • $\begingroup$ In the Bourbaki definition of "Coxeter group" (inspired mainly by work of Tits), a "reflection" always fixes a hyperplane, so your first sentence doesn't apply. For example, the identity element is not a "reflection". $\endgroup$
    – Jim Humphreys
    Commented Mar 19, 2017 at 16:17
  • $\begingroup$ You must understand that a hyperplane is space itself. Coxeter himself describes groups where there are order-three reflections etc, such as the group 3(3)3 => AAA = BBB = 1, ABA = BAB. Likewise, the intersection of two planes is an orthohedrix, the space orthogonal to a hedrix or 2-space. You will note then that (AB)^n constitutes a rotation. $\endgroup$
    – wendy.krieger
    Commented Mar 21, 2017 at 9:29

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