Last update: 7 May 2013
First we recall some definitions and facts from [Bou1968] and [BSa1972]. A group $W$ is called a Coxeter group (and the pair $(W,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ a Coxeter system), if it admits a presentation of the following form:
$$\begin{array}{cc}\text{(1.1)}& \text{generators:}\phantom{\rule{1em}{0ex}}{\sigma}_{1},\dots ,{\sigma}_{\ell}\text{.}\phantom{\rule{2em}{0ex}}\text{relations:}\phantom{\rule{1em}{0ex}}{({\sigma}_{i},{\sigma}_{j})}^{{m}_{ij}}=1,\hspace{0.17em}i,j\in \{1,\dots ,\ell \}\text{.}\end{array}$$where ${\left\{{m}_{ij}\right\}}_{i,h\in \{1,\dots ,\ell \}},$ with ${m}_{ij}\in N\cup \left\{\infty \right\}$ and ${m}_{ii}=1$ is the Coxeter matrix. In other words, $W$ is generated by (in the cases we are interested in, only finitely many) elements of order two; and by saying what the orders of the products of two such are, we have given a complete set of relations. The relations (1.1) are clearly equivalent to the following:
(1.2)
The Artin group (named this way and studied in [BSa1972]) belonging to $W$ (or to $\left\{{m}_{ij}\right\}\text{),}$ denoted by ${A}_{W},$ is the group we get by forgetting the relation (1.2)a. So it has a presentation:
$$\begin{array}{cc}\text{(1.3)}& \text{generators:}\phantom{\rule{1em}{0ex}}{s}_{1},\dots ,{s}_{\ell}\text{.}\phantom{\rule{2em}{0ex}}\text{relations:}\phantom{\rule{1em}{0ex}}{s}_{i}{s}_{j}{s}_{i}\dots ={s}_{j}{s}_{i}{s}_{j}\dots \phantom{\rule{1em}{0ex}}\text{(each side}\hspace{0.17em}{m}_{ij}\hspace{0.17em}\text{factors and}\hspace{0.17em}i\ne j\text{)}\end{array}$$The pair $({A}_{W\prime},\{{s}_{1},\dots ,{s}_{\ell}\})$ is called an Artin system. Note that we can suppose that ${m}_{ij}\ge 2$ for $i\ne j$ and from now on we do so. We have also the Coxeter graph. This is a graph, having for each $i\in \{1,\dots ,\ell \}$ a vertex; and two vertices are joint by an edge if ${m}_{ij}\ge 3$ and if ${m}_{ij}\ge 4,$ then the number ${m}_{ij}$ is written along the edge.
An Artin group or Coxeter matrix or graph is called of finite type if the associated Coxeter group is finite. The Coxeter graphs (and hence the groups) of finite type are classified ([Bou1968]).
(1.4) Example. Braid groups.
Artin groups arise as fundamental croups of certain regular orbit spaces associated with the Coxeter groups. Let us recall the classical example. The permutation group on $\ell +1$ elements, ${S}_{\ell +1}$ is generated by the interchanges ${\sigma}_{i}\u2254(i-1\hspace{0.17em}i),$ $i=1,\dots ,\ell \text{.}$ And it is well known that $({S}_{\ell +1},\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ is a Coxeter system. Then ${m}_{ij}=2$ if $|i-j|>1$ and ${m}_{ij}=3$ if $|i-j|=1\text{.}$ So the Coxeter graph looks like:
$$\begin{array}{c}\n\n\n\n\n\n\n\n\n\\ \text{(Type}\hspace{0.17em}{A}_{\ell}\text{)}\end{array}$$This group acts on ${\u2102}^{\ell +1}$ by permuting coordinates. This action is free, when we restrict it to points, where all the coordinates are different. Dividing out the action we get the space of unordered $\ell +1\text{-tuples}$ of different complex numbers. The fundamental group of this space (choosing for $(0,1,\dots ,\ell )\in {\u2102}^{\ell +1}$ as basepoint) can easily be visualized as the so-called braid group on $\ell +1\text{-strings}:$ ${B}_{\ell +1}\text{.}$
Intuitively it is clear that ${B}_{\ell +1}$ is generated by $\{{s}_{1},\dots ,{s}_{\ell}\}$ where ${s}_{i}$ is a braid of the form:
$$\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n0\ni-1\ni\n\u2113\n\n\\ \text{Figure 13.}\end{array}$$Also we obviously have the following relations:
$$\begin{array}{c}{s}_{i}{s}_{j}={s}_{j}{s}_{i}\phantom{\rule{2em}{0ex}}\text{if}\hspace{0.17em}|i-j|\ge 2\\ {s}_{i}{s}_{i+1}{s}_{i}={s}_{i+1}{s}_{i}{s}_{i+1}\phantom{\rule{2em}{0ex}}i=1,\dots ,\ell -1\end{array}$$ $$\begin{array}{cc}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n0\ni-1\ni\ni+1\n\u2113\nsi\nsi+1\nsi\n\n\n\n\n& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n0\ni-1\ni\ni+1\n\u2113\nsi+1\nsi\nsi+1\n\n\n\n\n\\ \multicolumn{2}{c}{\text{Figure 14.}}\end{array}$$Artin showed that $\{{s}_{1},\dots ,{s}_{\ell}\}$ together with this relations forms a presentation of ${B}_{\ell +1}\text{.}$ So in fact he proved, that the braid group is the Artin group associated with the permutation group. Brieskorn [Bri1971] proved a similar theorem for all finite Coxeter groups. The case of affine Coxeter groups is considered by Nguyen Viet Dung [Ngu1983]. In this chapter we want to state and prove a theorem giving a geometric interpretation of Artin groups as fundamental groups of certain regular orbit spaces of Coxeter groups, generalising these results. The natural context for this theorem is the theory of "linear Coxeter groups" of Vinberg, which we will discuss next.
In this paragraph we want to summarise some of the results of Vinberg [Vin1971]. He considers a set of reflections in the faces of an open polyhedral cone (cf.(V1)), such that the images of the cone under the elements of the group generated by the reflections, are mutually disjoint (cf.(V2)). To be more exact:
(2.1) Definition. Let $\mathbb{V}$ be a finite dimensional real vector space. A pair $(C,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ is called a Vinberg pair if ${\sigma}_{i}\in \text{Gl}\left(\mathbb{V}\right)$ is a reflection in a hyperplane ${M}_{i}$ for $i=1,\dots ,\ell $ and $C\in K\left(\{{M}_{1},\dots ,{M}_{\ell}\}\right),$ satisfying the following two conditions:
(V1) | The polyhedral cone property: ${\mathcal{M}}^{C}=\{{M}_{1},\dots ,{M}_{\ell}\}$ |
(V2) | The Tits property: $w\left(C\right)\cap C\ne \varnothing \Rightarrow w={\text{id}}_{\mathbb{V}}$ for all $w\in W,$ where $W\subset \text{Gl}\left(\mathbb{V}\right)$ is the group generated by $\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\}\text{.}$ |
(2.2) Theorem (Vinberg). Let $(C,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ be a Vinberg pair, then
(a) | The set $I\u2254\bigcup _{w\in W}w\left(\stackrel{\u203e}{C}\right)$ is a convex cone and $W$ acts properly discontinuously on the interior $\stackrel{\circ}{I}\text{.}$ |
(b) | If $x\in \stackrel{\u203e}{C},$ then the stabiliser of $x$ is the group generated by $\left\{{\sigma}_{i}\hspace{0.17em}\right|\hspace{0.17em}x\in {M}_{i}\}$ and $x\in \stackrel{\circ}{I}$ iff this group is finite. |
(c) | The pair $(W,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ is a Coxeter system. |
Proof. | |
See Vinberg [Vin1971] theorem 2, page 1092. $\square $ |
(2.3) Definition. A Coxeter system (respectively Coxeter group) as obtained from a Vinberg pair by (2.2)(c) is called a linear (or geometric) Coxeter system (respectively group). We often will suppose, that a Vinberg pair is fixed.
(2.4) Corollary. The interior $\stackrel{\circ}{I}$ is exactly the set of points in $I$ with finite stabiliser.
(2.5) Corollary. We have $I=\mathbb{V}$ iff $W$ is finite.
(2.6) Corollary. Let $\mathcal{M}$ denote the set of reflection hyperplanes of reflections of $W$ (the "mirrors" of $W\text{).}$ Then $\mathcal{M}$ is locally finite on $\stackrel{\circ}{I}\text{.}$ $(V,\stackrel{\circ}{I},\mathcal{M})$ is a hyperplane configuration (cf.(I.2.6)).
Proof of (2.6). | |
Since the action of $W$ on $\stackrel{\circ}{I}$ is properly discontinuous, a point $x\in \stackrel{\circ}{I}$ has a neighbourhood of $U$ such that $w\left(U\right)\cap U\ne \varnothing \Rightarrow w\in {W}_{x},$ the stabiliser of $x\text{.}$ So if $U\cap M\ne \varnothing ,$ $M\in \mathcal{M},$ then the reflection in $M$ is in ${W}_{x},$ which is finite. $\square $ |
Vinberg also gives a complete theory how to construct Vinberg pairs (and so linear Coxeter groups). One consequence of this is the existence of at least one linear Coxeter group isomorphic to a given abstract Coxeter group. So the geometric interpretation of Artin groups given in the next paragraph applies to all of them. Another consequence is an easy way to introduce the so-called extended Coxeter (or Weyl) groups in chapter III. So we briefly describe these results. First a result from linear algebra. Let $\mathbb{V}$ be a linear space and ${\mathbb{V}}^{*}$ its dual. For svstems ${v}_{1},\dots ,{v}_{\ell}\in \mathbb{V},$ ${\varphi}_{i},\dots ,{\varphi}_{\ell}\in {\mathbb{V}}^{*},$ we define the following invariants:
(2.7)
The set $\{A,{L}_{v},{L}_{\varphi},d\}$ will be called the characteristic of the system $\{{v}_{i},{\varphi}_{i}\}\text{.}$ The space ${L}_{v}$ (respectively ${L}_{\varphi}\text{)}$ is a subspace of ${L}_{c}\left(A\right)$ (respectively ${L}_{r}\left(A\right)\text{),}$ the space of linear relations among the columns (respectively rows) of $A\text{.}$ For if $({x}_{1},\dots ,{x}_{\ell})\in {L}_{v},$ then for each $i\in \{1,\dots ,\ell \}$ we have ${x}_{1}{a}_{i1}+\dots +{x}_{\ell}{a}_{i\ell}={x}_{1}{\varphi}_{i}\left({v}_{1}\right)+\dots +{x}_{\ell}{\varphi}_{i}\left({v}_{\ell}\right)={\varphi}_{i}({x}_{1}{v}_{1}+\dots +{x}_{\ell}{v}_{\ell})={\varphi}_{i}\left(0\right)=0\text{.}$ Let ${d}_{v}$ denote the codimension of ${L}_{v}$ in ${L}_{c}\left(A\right),$ and ${d}_{\varphi}$ the codimension of ${L}_{\varphi}$ in ${L}_{r}\left(A\right)\text{.}$ Such a system is called isomorphic to another $\{{v}_{i}^{\prime}\in \mathbb{V}\prime ,{\varphi}_{1}^{\prime}\in {\left(\mathbb{V}\prime \right)}^{*},i=1,\dots ,\ell \},$ if there exists an isomorphism from $\mathbb{V}$ to $\mathbb{V}\prime $ transforming ${v}_{i}$ into ${v}_{i}^{\prime}$ and ${\varphi}_{i}^{\prime}$ into ${\varphi}_{i}\text{.}$
(2.8) Proposition. Let $A$ be any $\ell \times \ell \text{-matrix,}$ let ${L}_{v}$ and ${L}_{\varphi}$ be subspaces of ${L}_{c}\left(A\right)$ and ${L}_{r}\left(A\right)$ respectively, and let $d$ be a nonnegative integer. Then there exists a system $\{{v}_{i}\in \mathbb{V},{\varphi}_{i}\in {\mathbb{V}}^{*},i=1,\dots ,\ell \}$ unique up to isomorphism, having $\{A,{L}_{v},{L}_{\varphi},d\}$ as its characteristic. Also:
(2.9) $\text{dim}\left(\mathbb{V}\right)=\text{rank}\left(A\right)+{d}_{v}+{d}_{\varphi}+d\text{.}$
Proof. | |
See Vinberg [Vin1971] prop. 15. $\square $ |
Let ${\sigma}_{i}$ be the linear map given by ${\sigma}_{i}\left(x\right)\u2254x-{\varphi}_{i}\left(x\right){v}_{i}$ for $x\in \mathbb{V}\text{.}$ If ${\sigma}_{i}$ is a reflection, then ${a}_{ii}\ne 0$ and ${v}_{i}$ has to be the eigenvector with eigenvalue $-1,$ so then one has ${a}_{ii}=2,$ hence the necessity of the first part of condition (C2) below.
Let $C\u2254\{x\in \mathbb{V}\hspace{0.17em}|\hspace{0.17em}{\varphi}_{i}\left(x\right)>0,i=1,\dots ,\ell \}\text{.}$ If $C\ne \varnothing ,$ say $c\in C,$ then if ${x}_{1}{\varphi}_{1}\left(c\right)+\dots +{x}_{\ell}{\varphi}_{\ell}\left(c\right)=0,$ then all ${x}_{1},\dots ,{x}_{\ell}$ are zero or at least one is negative. So we have seen why (C2) below is necessary.
(2.10) Proposition Vinberg. Let the system $\{{v}_{i}\in \mathbb{V},{\varphi}_{i}\in {\mathbb{V}}^{*},i=1,\dots ,\ell \}$ have characteristic $\{A,{L}_{v},{L}_{\varphi},d\}\text{.}$ Let $C$ and ${\sigma}_{1},\dots ,{\sigma}_{\ell}$ be as above. Then $(C,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ is a Vinberg pair if and only if the following conditions are satisfied:
(C1) | ${a}_{ij}\le 0$ for $i\ne j$ and if ${a}_{ij}=0,$ then ${a}_{ji}=0\text{.}$ |
(C2) | ${a}_{ii}=2$ for $i=1,\dots ,\ell $ and ${a}_{ij}{a}_{ji}\ge 4$ or ${a}_{ij}{a}_{ji}=4{\text{cos}}^{2}(\pi /{m}_{ij})$ for a positive integer ${m}_{ij}\text{.}$ |
(C3) | ${L}_{\varphi}\cap \{({x}_{1},\dots ,{x}_{\ell})\in {L}_{r}\left(A\right)\hspace{0.17em}|\hspace{0.17em}{x}_{i}\ge 0,i=1,\dots ,\ell \}=\left\{0\right\}\text{.}$ |
Agreeing that ${m}_{ij}=\infty ,$ if ${a}_{ij}{a}_{ji}\ge 4,$ then ${\left\{{m}_{ij}\right\}}_{i,j=1,\dots ,\ell}$ is exactly the Coxeter matrix of the Coxeter system $(W,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})\text{.}$
Proof. | |
See Vinberg [Vin1971] theorem 5. p 1105. $\square $ |
(2.11) Remarks. The polyhedral cone property (2.1)(V1) follows from (C1), (C3) and ${a}_{ii}\ne 0$ for $i=1,\dots ,\ell \text{.}$
If the polyhedral cone property holds, then (C1) and (C2) is equivalent to the Tits property (2.1)(V2).
Suppose we have a Cartan matrix satisfying (C1) and (C2). By choosing ${L}_{\varphi}=\left\{0\right\}$ we can always force (C3) to be true. In this case $C$ is even a simplicial cone.
Now let $M={\left\{{m}_{ij}\right\}}_{i,j=1,\dots ,\ell}$ be an arbitrary Coxeter matrix. Then the matrix $\text{Cos}\left(M\right),$ which has by definition entries $-2\text{cos}(\pi /{m}_{ij})$ satisfies (C1) and (C2) (if we agree, that $\pi /\infty =0\text{).}$ The linear Coxeter group with characteristic $(\text{Cos}\left(M\right),{L}_{v}(\text{Cos}\left(M\right),0,0))$ is called the Tits (linear) representation of the abstract Coxeter group associated to $M\text{.}$ This linear Coxeter group is reduced (cf. [Vin1971]). Note that $\text{dim}\left({L}_{v}\left(\text{Cos}\left(M\right)\right)\right)=\ell -\text{rank}\left(\text{Cos}\left(M\right)\right),$ so in this case $\text{dim}\left(\mathbb{V}\right)=\ell \text{.}$ (cf. Bourbaki [Bou1968] Ch.V §4.4 "La representation contraorediante"). So we nave:
(2.12) Corollary. Each abstract Coseter group is isomorphic to a linear Coxeter group.
(2.13) Finally let $N={\left\{{n}_{ij}\right\}}_{i,j=1,\dots ,\ell}$ be a Cartan matrix satisfying (C1) and ${n}_{ii}=2$ with integer entries. Then (C2) is automatically satisfied with ${m}_{ij}=2,3,4,6$ or $\infty $ as ${n}_{ij}{n}_{ji}=0,1,2,3,$ or $\ge 4$ respectively. Choosing ${L}_{v}=0$ and ${L}_{\varphi}=0$ we get a linear Coxeter group acting on a lattice $Q\u2254\underset{i=1}{\overset{\ell}{\u2a01}}\mathbb{Z}{v}_{i}\text{.}$ This leads to the notion of an extended Coxeter group and is connected with the notion of a generaliserd root system. This situation will be treated in chapter III.
After this exposition of the theory of Vinberg we conclude this paragraph by studying the Tits galleries of the hyperplane configuration $(\mathbb{V},\stackrel{\circ}{I},\mathcal{M})$ associated to a linear Coxeter group of a Vinberg pair.
(2.14) Lemma. There is a bijective correspondence between the set of conjugates of the generators ${\sigma}_{i}$ in $W$ and the set of mirrors of $W$ $\left(\mathcal{M}\right)\text{.}$
Proof. | |
In general each element $M\in \mathcal{M}$ is a wall of some chamber. For a point $x\in M$ in the cone $\stackrel{\circ}{I}$ has, by the locally finiteness, a neighbourhood meeting finitely many facets inside $M\text{.}$ At least one of these has the same dimension as M, hence is a face of some chamber. In this case this implies that $M$ is of the form $w\left({M}_{i}\right)$ for some $w\in W$ and $i\in \{1,\dots ,\ell \}\text{.}$ So if we know that for all $w,v\in W$ and $i,j\in \{1,\dots ,\ell \}$ we have: $$\begin{array}{cc}\text{(2.15)}& w{\sigma}_{i}{w}^{-1}=v{\sigma}_{j}{v}^{-1}\iff w\left({M}_{i}\right)=v\left({M}_{j}\right)\end{array}$$then $w{\sigma}_{i}{w}^{-1}\to w\left({M}_{i}\right)$ sets up the desired bijective correspondence. Now (2.15) is equivalent to $$\begin{array}{cc}\text{(\$)}& {\sigma}_{i}=w{\sigma}_{j}{w}^{-1}\leftrightarrow {M}_{i}=w\left({M}_{j}\right)\phantom{\rule{1em}{0ex}}\text{for}\hspace{0.17em}w\in W\hspace{0.17em}\text{and}\hspace{0.17em}i,j\in \{1,\dots ,\ell \}\end{array}$$To prove ($) note that $$\begin{array}{cc}{\sigma}_{i}& \text{is a reflection with reflection hyperplane}\hspace{0.17em}{M}_{i}\\ w{\sigma}_{j}{w}^{-1}& \text{is a reflection with reflection hyperplane}\hspace{0.17em}w\left({M}_{j}\right)\text{.}\end{array}$$So if ${\sigma}_{i}=w{\sigma}_{j}{w}^{-1},$ then ${M}_{i}=w\left({M}_{j}\right)\text{.}$ On the other hand if ${M}_{i}=w\left({M}_{j}\right),$ then ${\sigma}_{i}$ and $w{\sigma}_{j}{w}^{-1}$ are both reflections fixing ${M}_{i}\cap \stackrel{\u203e}{C},$ which is the closure of some face of $C$ and the stabiliser of this face is the group $\{{\text{id}}_{\mathbb{V}},{\sigma}_{i}\}$ (by theorem (2.2)(b)). So then $w{\sigma}_{j}{w}^{-1}={\sigma}_{i}\text{.}$ $\square $ |
(2.16) Proposition. For each sequence ${i}_{1},\dots ,{i}_{k}\in \{1,\dots ,\ell \}$ the sequence:
$$\begin{array}{cc}\text{(2.17)}& C,{C}_{1},\dots ,{C}_{k}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{C}_{r}={\sigma}_{{i}_{1}}{\sigma}_{{i}_{2}}\dots {\sigma}_{{i}_{r}}\left(C\right)\phantom{\rule{1em}{0ex}}r=1,\dots ,k\end{array}$$is a Tits gallery and each Tits gallery starting in $C$ is obtainable this way from a unique sequence ${i}_{1},\dots ,{i}_{k}\text{.}$
This Tits gallery is direct if and only if the length of ${\sigma}_{{i}_{1}}{\sigma}_{{i}_{2}}\dots {\sigma}_{{i}_{k}}$ as element of the Coxeter group is equal to $k\text{.}$
Proof. | |
We have $\mathcal{M}({C}_{r-1},{C}_{r})=\mathcal{M}({w}_{r}\left(C\right),{w}_{r}{\sigma}_{{i}_{r}}\left(C\right))={w}_{r}\left(\mathcal{M}(C,{\sigma}_{{i}_{r}}\left(C\right))\right)=\left\{{w}_{r}\left({M}_{{i}_{r}}\right)\right\}$ where ${w}_{r}\u2254{\sigma}_{{i}_{1}}{\sigma}_{{i}_{2}}\dots {\sigma}_{{i}_{r-1}}\text{.}$ So $\left|\mathcal{M}({C}_{r-1},{C}_{r})\right|=1$ and the sequence (2.17) is indeed a Tits gallery. Moreover we see that ${M}_{{i}_{1}}={w}_{1}\left({M}_{{i}_{1}}\right),{w}_{2}\left({M}_{{i}_{2}}\right),\dots ,{w}_{k}\left({M}_{{i}_{k}}\right),$ is its sequence of mirrors. According to the definition (see (1.3.21) and (1.3.20)(a)) this Tits gallery is direct iff all the mirrors are different. Lemma (2.14) tells us that this is the case iff all the conjugates ${w}_{1}{\sigma}_{{i}_{1}}{w}_{1}^{-1},\dots ,{w}_{k}{\sigma}_{{i}_{k}}{w}_{k}^{-1}$ are different. By Bourbaki [Bou1968] Ch. IV §1.4 lemme 2 this in turn is equivalent to the condition in the proposition. Let $C,{C}_{1},\dots ,{C}_{k}$ be a Tits gallery and suppose by induction that we have proved, that there is a unique sequence ${i}_{1},\dots ,{i}_{k-1}$ such that ${C}_{r}$ is given as in (2.17) for $r=1,\dots ,k-1\text{.}$ The chambers ${C}_{k-1}$ and ${C}_{k}$ are separated by exactly one common wall, so also $C={w}_{k-1}^{-1}\left({C}_{k-1}\right)$ and ${w}_{k-1}^{-1}\left({C}_{k}\right),$ say by ${M}_{{i}_{k}}\text{.}$ Note that ${i}_{k}$ is unique. Then ${w}_{k-1}^{-1}\left({C}_{k}\right)={\sigma}_{{i}_{k}}\left(C\right),$ so ${C}_{k}={w}_{k-1}{\sigma}_{{i}_{k}}\left(C\right)={\sigma}_{{i}_{1}}\dots {\sigma}_{{i}_{k}}\left(C\right)\text{.}$ $\square $ |
(2.18) Example. If $i\ne j$ and $k\le {m}_{ij},$ then ${\sigma}_{i}{\sigma}_{j}{\sigma}_{i}\dots $ $\text{(}k$ factors) has length $k,$ as is well known.
Let $(W,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ be a linear Coxeter system (2.3) in a space $\mathbb{V}$ and let ${\varphi}_{i},C,{M}_{i},I,\stackrel{\circ}{I}$ and $\mathcal{M}$ be as in §2. The group $W$ acts also properly discontinuously on the domain:
$\begin{array}{cc}\text{(3.1)}& \Omega \u2254{\text{Im}}^{-1}\left(\stackrel{\circ}{I}\right)\subset {\mathbb{V}}_{\u2102}\end{array}$The regular points of the action, which are by definition the points with trivial stabiliser are the points of the space:
$\begin{array}{cc}\text{(3.2)}& Y\u2254\Omega -\text{{reflecetion hyperplanes}}=\Omega -\bigcup _{M\in \mathcal{M}}{M}_{\u2102}\text{.}\end{array}$Now the action of $W$ on $Y$ is proper and free, so we get a space
$\begin{array}{cc}\text{(3.3)}& X\u2254Y/W\end{array}$which is a complex manifold. This space is called the regular orbit space associated to the linear Coxeter group.
(3.4) Let $p:Y\to X$ denote the canonical projection. Choose a point
$$\begin{array}{cc}\text{(3.5)}& c\in $ C\text{.}$\end{array}$$Let $\hat{*}\u2254\sqrt{-1}c$ and $*\u2254p\left(\hat{*}\right)$ as base point for $X\text{.}$ For $i=1,\dots ,\ell $ define the curve ${\mathcal{S}}_{i}$ in $Y$ from $\hat{*}$ to ${\sigma}_{i}\left(\hat{*}\right)$ by:
$$\begin{array}{cc}\text{(3.6)}& {\mathcal{S}}_{i}\left(t\right)\u2254v\left(t\right)+\sqrt{-1}((1-t)c+t{\sigma}_{i}\left(c\right))\end{array}$$where $v:[0,1]\to \mathbb{V}$ is continuous and $v\left(1\right)=v\left(0\right)=0$ and ${\varphi}_{i}\left(v(?)\right)>0\text{.}$
(3.7) Finally let ${s}_{i}\u2254\{p\circ {\mathcal{S}}_{i}\}\in \pi (X,*)\text{.}$
Now we can state and prove the main theorem of this chapter:
(3.8) Theorem. The fundamental group of a regular orbit space associated to a linear Coxeter system is the Artin group associated to the abstract Coxeter system. More precisely, with the notations above:
The group $\pi (X,*)$ is generated by ${s}_{1},\dots ,{s}_{\ell}$ and a complete set of relations is:
$$\begin{array}{cc}\text{(3.9)}& {s}_{i}{s}_{j}{s}_{i}\dots ={s}_{j}{s}_{i}{s}_{j}\dots \end{array}$$where ${m}_{ij}<\infty $ and each side has ${m}_{ij}$ factors.
For the proof we need some more preparation, which is also of general use. First we discuss the effects of a group acting on a general hyperplane system. The assertions are clear from naturality and also exact proofs are not difficult.
(3.10) Discussion. Let $(\mathbb{V},H,\mathcal{M},p)$ be a hyperplane system. Let $W\subset \text{Gl}\left(\mathbb{V}\right)$ be a group with the property that for all $w\in W,$ $M\in \mathcal{M}$ and $N\in {p}_{M}$ we have:
$$\begin{array}{cc}\text{(3.11)}& w\left(M\right)\in \mathcal{M}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}w\left(N\right)\in {p}_{w\left(M\right)}\text{.}\end{array}$$Then there is a functorial action of $W$ on $\text{Pgal}$ and $\text{Gal}\text{.}$ This means that $W$ permutes $K\left(\mathcal{M}\right)$ and if $G\in {\text{Pgal}}_{AB}$ (or ${\text{Gal}}_{AB}\text{),}$ then $w\left(G\right)\in {\text{Pgal}}_{w\left(A\right)w\left(B\right)}$ (respectively ${\text{Pgal}}_{w\left(A\right)w\left(B\right)}\text{)}$ and $w\left({G}_{1}{G}_{2}\right)=w\left({G}_{1}\right)w\left({G}_{2}\right)\text{.}$
Suppose we are given in each chamber $C\in K\left(\mathcal{M}\right)$ a point ${a}_{C}$ such that $w\left({a}_{C}\right)={a}_{w\left(C\right)}$ for all $w\in W$ and $C\in K\left(\mathcal{M}\right)\text{.}$ Then the subgroupoid $\Pi $ of the fundamental groupoid of the space $Y(\mathbb{V},H,\mathcal{M},p)$ is defined and the functor $\phi :\text{Pgal}$ (respectively $\text{Gal}\text{)}\to \Pi ,$ as occuring in thm. (I.2.21) (respectively (I.4.10)) is equivariant under the action of $W$ on $\text{Pgal}$ (respectively $\text{Gal}\text{)}$ and the natural action of $W$ on $\Pi \text{.}$
(3.12) Proposition. If moreover $W$ permutes $K\left(\mathcal{M}\right)$ simply transitively, then $\text{Gal}/W$ is a semigroup freely generated by the $W\text{-orbits}$ of the following one step galleries:
$$\begin{array}{cc}\text{(3.13)}& {C}^{{\textstyle U}}{\text{sp}}_{M}\left(C\right)\end{array}$$for some fixed chamber $C\in K\left(\mathcal{M}\right)$ and all $M\in {\mathcal{M}}^{C}$ and $U\in K\left({p}_{M}\right)\text{.}$
Proof. | |
If $W$ acts transitively on $K\left(\mathcal{M}\right),$ $\text{Gal}/W$ is a category with only one object, hence a semigroup. Now fix $C\in K\left(\mathcal{M}\right)\text{.}$ If the action is simply transitive, then each one step gallery ${A}^{{\textstyle U}}B$ is in the $W\text{-orbit}$ of a unique one step gallery of the form (3.13). Since $\text{Gal}$ is by definition freely generated by the one step galleries the freeness for $\text{Gal}/W$ follows too. $\square $ |
A linear Coxeter group satisfies (3.11) and the condition of (3.12), so we have:
(3.14) Corollary. If $(\mathbb{V},\stackrel{\circ}{I},\mathcal{M})$ is the simple hyperplane system of a linear Coxeter group $W,$ then $\text{Gal}/W$ is a free semigroup, freely generated by ${S}_{1}^{+},\dots ,{S}_{\ell}^{+},{S}_{1}^{-},\dots ,{S}_{\ell}^{-}$ where ${S}_{i}^{\epsilon}$ is the $W\text{-orbit}$ of the one step gallery:
$${C}^{{\textstyle \epsilon}}{\sigma}_{i}\left(C\right)\phantom{\rule{2em}{0ex}}\epsilon \in \{+,-\},\phantom{\rule{2em}{0ex}}i\in \{1,\dots ,\ell \}$$Furthermore a gallery ${C}^{{\textstyle {\epsilon}_{1}}}{{C}_{1}}^{{\textstyle {\epsilon}_{2}}}\dots {}^{{\textstyle {\epsilon}_{k}}}{C}_{k}$ is in the $W\text{-orbit}$
$$\begin{array}{cc}\text{(3.15)}& {{S}_{{i}_{1}}}^{{\textstyle {\delta}_{1}}}{{S}_{{i}_{2}}}^{{\textstyle {\delta}_{2}}}\dots {{S}_{{i}_{m}}}^{{\textstyle {\delta}_{m}}}\in \text{Gal}/W\end{array}$$if and only if $k=m,$ ${\epsilon}_{i}={\delta}_{i}$ all $i-1,\dots ,k$ and for $r=1,\dots ,k$
$${C}_{r}={\sigma}_{{i}_{1}}{\sigma}_{{i}_{2}}\dots {\sigma}_{{i}_{r}}\left(C\right)\text{.}$$If ${\delta}_{1}=\dots ={\delta}_{m}=+$ and ${\sigma}_{{i}_{1}}\dots {\sigma}_{{i}_{m}}$ has length $m$ in the Coxeter group, then the galleries in the $W\text{-orbit}$ (3.15) are direct.
Proof. | |
To prove equality of the signs we note that if ${A}^{{\textstyle \epsilon}}B$ is a one-step gallery and $w\in W$ then $w({A}^{{\textstyle \epsilon}}\mathrm{B)}={w\left(A\right)}^{{\textstyle \epsilon}}w\left(B\right)\text{.}$ For if $\mathcal{M}(A,B)=\left\{M\right\},$ then for instance $(\epsilon =+)$ $w\left({D}_{M}\left(A\right)\right)={D}_{w\left(M\right)}\left(w\left(A\right)\right)\text{.}$ The rest of the assertions follow easily from (2.16), (3.10) and (3.12). $\square $ |
Proof of theorem (3.8). | |
First note that the the space $Y$ is the same as the space $Y(\mathbb{V},\stackrel{\circ}{I},\mathcal{M})$ as defined in (1.2.9). We have already the point $c\in C$ (3.5) and the group $W$ satisfies the conditions of the discussion (3.10) and proposition (3.12). So the groupoid $\Pi $ is defined and the projection $p:Y\to X$ induces a functor ${p}_{*}:\Pi \to \pi (X,*)\text{.}$ From the fact that ${p}^{-1}(*)=\left\{\sqrt{-1}w\left(c\right)\hspace{0.17em}\right|\hspace{0.17em}w\in W\}$ and the unique path lifting property it follows that ${p}_{*}$ is surjective and it is also easy to see that ${p}_{*}\left(\lambda \right)={p}_{*}\left(\mu \right)$ iff $\lambda =w\left(\mu \right)$ for some $w\in W\text{.}$ So we have a natural (3.16) isomorphism: ${p}_{*}/W:\Pi /W\to \pi (X,*)$ According to the discussion.in (3.10) the map $\phi :\text{Gal}\to \Pi $ is equivariant for the action of $W$ on $\text{Gal}$ and $\Pi ,$ so we have a map $\phi /W:\text{Gal}/W\to \Pi /W\text{.}$ Now $\phi /W$ is surjective, for $\phi $ is so and $\phi /W$ is a homomorphism of semigroups because $\phi $ is a functor. From the equivariance it follows that the equivalence relation on $\text{Gal}/W$ "having the same $\phi /W\text{-image"}$ is generated by the $W\text{-orbits}$ of the pairs, which generate the gallery equivalence relation. By remark (1.5.4) for these we can take the cancel relations and the positive flip relations. If we consider the $W\text{-orbits}$ of the cancel relations (I.5.2) $({A}^{{\textstyle +}}{B}^{{\textstyle -}}A,{\varnothing}_{A})$ and $({A}^{{\textstyle -}}{B}^{{\textstyle +}}A,{\varnothing}_{A})$ we see that we get all of them by taking $A=C$ and $B={\sigma}_{i}\left(C\right)$ $i=1,\dots ,\ell ,$ so they are the $W\text{-orbits}$ of $({C}^{{\textstyle +}}{{\sigma}_{i}\left(C\right)}^{{\textstyle -}}C,{\varnothing}_{C})$ and $({C}^{{\textstyle -}}{{\sigma}_{i}\left(C\right)}^{{\textstyle +}}C,{\varnothing}_{C})$ so we get the pairs: $$\begin{array}{cc}\text{(3.17)}& ({S}_{i}^{+}{S}_{i}^{-},\varnothing )\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}({S}_{i}^{-}{S}_{i}^{+},\varnothing )\phantom{\rule{2em}{0ex}}i=1,\dots ,\ell \end{array}$$Now we consider the $W\text{-orbits}$ of the positive flip relations Again we can suppose ${C}_{0}={D}_{0}=C$ now formula (I.5.3). If $P$ is a plinth of $C,$ then since $\text{dim}\left(P\right)=2$ there are two walls of $C$ say ${M}_{i}$ and ${M}_{j}$ such that ${M}_{i}\cap {M}_{j}$ is the support of $P\text{.}$ The group generated by ${\sigma}_{i}$ and ${\sigma}_{j}$ stabilises $P\subset \stackrel{\circ}{I}$ so it is finite (theorem (2.2)(b)), hence ${m}_{ij}<\infty $ and it follows from proposition (2.16) that ${C}_{0},{C}_{1},..,{C}_{{m}_{ij}}$ and ${D}_{0},{D}_{1},\dots ,{D}_{{m}_{ij}},$ where ${C}_{r}\u2254{w}_{r}\left(C\right)$ with ${w}_{r}\u2254{\sigma}_{i}{\sigma}_{j}{\sigma}_{i}\dots $ $\text{(}r$ factors) and ${D}_{r}\u2254{w}_{r}^{\prime}\left(C\right)$ with ${w}_{r}^{\prime}\u2254{\sigma}_{j}{\sigma}_{i}{\sigma}_{j}\dots $ $\text{(}r$ factors) are two, hence the two direct (by example (2.18)) Tits galleries from $C$ to ${\text{sp}}_{P}\left(C\right)\text{.}$ On the other hand if $i,j\in \{1,\dots ,\ell \}$ and ${m}_{ij}<\infty ,$ then the set $P\u2254\{x\in W\hspace{0.17em}|\hspace{0.17em}{\varphi}_{r}\left(x\right)>0\hspace{0.17em}\text{if}\hspace{0.17em}r\ne i,j\hspace{0.17em}\text{and}\hspace{0.17em}{\varphi}_{i}\left(x\right)={\varphi}_{j}\left(x\right)=0\}\subset \stackrel{\u203e}{C}$ is nonempty as follows from Vinberg [Vin1971], theorem 7. pg 1114. Its $W\text{-stabiliser}$ is finite since ${m}_{ij}<\infty ,$ so $P\subset \stackrel{\circ}{I}$ and $P$ is a plinth of $C\text{.}$
It follows that the $W\text{-orbits}$ of the positive flip relations are: Summarising we have proved the following: We have a surjective homomorphism from the free semigroup $\text{Gal}/W$ freely generated by ${S}_{1}^{+},\dots ,{S}_{\ell}^{+},{S}_{1}^{-},\dots ,{S}_{\ell}^{-}$ to the group $\pi (X,*)$ $\text{(}\phi /W$ and the natural isomorphism ${p}_{*}/W\text{),}$ sending ${S}_{i}^{+}$ to ${s}_{i}$ and ${S}_{i}^{-}$ to ${s}_{i}^{-1}$ and the images of two are equal iff they are equivalent by means of the cancel relations (3.17) and the flip relations (3.18). This proves the theorem. $\square $ |
(3.19) Remark. Two words in the elements ${\sigma}_{i}$ are minimal representatives of the same element in $W$ if and only if they are equivalent by means of the relations ${\sigma}_{i}{\sigma}_{j}{\sigma}_{i}\dots ={\sigma}_{j}{\sigma}_{i}{\sigma}_{j}\dots $ (both sides ${m}_{ij}<\infty $ factors). This follows from (I.3.26) and (2.16). It implies that ${\sigma}_{i}\to {s}_{i}$ induces a map from $W$ to ${A}_{W},$ which is injective since the projection ${A}_{W}\to W$ is a left inverse. But the map is not a homomorphism!
With the help of the geometrical interpretation of Artin groups we shall prove some of their algebraic properties. For instance, we show in this way that the subgroup generated by a subset of the generators forms an Artin system. And the intersection of two such groups is the group generated by the common generators of the two. These facts were proved for so called Artin groups of large type (which means that ${m}_{ij}\ge 3$ for $i\ne j,$ so there are no commution relations ${m}_{ij}=2\text{)}$ by Appel [ASc1983]. First some preparations. Let $W$ be a linear Coxeter group with Vinberg pair $(C,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})\text{.}$
(4.1) Proposition. (cf. Bourbaki [Bou1968] Ch.V §4.6) The closure $\stackrel{\u203e}{C}$ is a fundamental domain for the action of $W$ on $I$ in the following strong sense: if $x\in \stackrel{\u203e}{C}$ and $w\in W$ such that $w\left(x\right)\in \stackrel{\u203e}{C},$ then $w\left(x\right)=x\text{.}$
Proof. | |
Since $w\left(x\right)\in \stackrel{\u203e}{C}\cap w\left(\stackrel{\u203e}{C}\right),$ we see that if $w\left(x\right)\notin M\in \mathcal{M},$ then $\stackrel{\u203e}{C}$ and $w\left(\stackrel{\u203e}{C}\right)$ meet, hence are contained in $\stackrel{\u203e}{{D}_{M}\left(w\left(x\right)\right)}\text{.}$ So $M\notin \mathcal{M}(C,w\left(C\right))\text{.}$ The element $w$ can be written as a product of reflections in the hyperplanes separating $C$ and $w\left(C\right)$ and these all contain $w\left(x\right)$ as we saw. Hence $w\left(x\right)=x\text{.}$ $\square $ |
(4.2) Definition. Let $J\subset \{1,\dots ,\ell \}$ and suppose that $({C}_{J},\left\{{\sigma}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in J\})$ is a Vinberg pair for some ${C}_{J},$ then we denote the linear Coxeter group, the set of mirrors, the Tits cone, the space of regular points, the regular orbit space and the Artin group by ${W}_{J},{\mathcal{M}}_{J},{I}_{J},{Y}_{J},{X}_{J}$ and ${A}_{{W}_{J}}$ respectively. We call a subset $J\subset \{1,\dots ,\ell \}$ of facettype (w.r.t. a facet $F\text{),}$ if $\bigcap _{j\in J}{M}_{j}={L}_{F},$ the support of $F,$ where the facet $F$ is in $\stackrel{\u203e}{C}\text{.}$ For instance for the Tits representation all subsets of $\{1,\dots ,\ell \}$ are of facettype.
(4.3) Remark. In fact for each $J\subset \{1,\dots ,\ell \}$ there is such a ${C}_{J}$ (cf. Vinberg [Vin1971] Cor. 3 pg. 1105). This follows from theorem (2.10). We need the fact only in the case that $J$ is of facettype and then the proof is simple and will be given (4.6)(i).
(4.4) Definition. Let $J\subset \{1,\dots ,\ell \}$ and let $H$ be a left coset of ${W}_{J}\text{.}$ Then we define the star of $H$ by $\text{Star}\left(H\right)\u2254\bigcup _{w\in H}w\left({C}^{J}\right),$ where ${C}^{J}$ is defined as ${C}^{J}\u2254\{x\in \stackrel{\u203e}{C}\hspace{0.17em}|\hspace{0.17em}x\in {M}_{i}\Rightarrow i\in J\}\text{.}$ In particular we define $\text{St}\left(J\right)\u2254\text{Star}\left({W}_{J}\right)\text{.}$ Note that $\text{Star}(w.{W}_{J})=w\left(\text{St}\left(J\right)\right)\text{.}$
(4.5) Proposition.
(i) | If $J\prime \subset J,$ then ${C}^{J\prime}\subset {C}^{J}$ and $\text{St}\left(J\prime \right)\subset \text{St}\left(J\right)\text{.}$ |
(ii) | ${C}^{{J}_{1}}\cap {C}^{{J}_{2}}={C}^{{J}_{1}\cap {J}_{2}}$ and $\text{St}\left({J}_{1}\right)\cap \text{St}\left({J}_{2}\right)=\text{St}({J}_{1}\cap {J}_{2}),$ ${J}_{i}\subset \{1,\dots ,\ell \}\text{.}$ |
(iii) | $\text{Star}\left(H\right)$ is convex. |
Proof. | |
The only not obvious fact of (i) and (ii) is $\text{St}\left({J}_{1}\right)\cap \text{St}\left({J}_{2}\right)\subset \text{St}({J}_{1}\cap {J}_{2})\text{.}$
For this suppose
$x\in {w}_{1}\left({C}^{{J}_{1}}\right)\cap {w}_{2}\left({C}^{{J}_{2}}\right)\subset {w}_{1}\left(\stackrel{\u203e}{C}\right)\cap {w}_{2}\left(\stackrel{\u203e}{C}\right),$
with ${w}_{i}\in {W}_{{J}_{i}}\text{.}$ Since
${w}_{1}^{-1}\left(x\right)\in \stackrel{\u203e}{C}\cap {w}_{1}^{-1}{w}_{2}\left(\stackrel{\u203e}{C}\right)$
we have by (4.1) that (iii) is proved analogously as in Bourbaki [Bou1968] Ch. V §4.6 prop. 6(i). $\square $ |
(4.6) Proposition. Suppose $(C,\{{\sigma}_{1},\dots ,{\sigma}_{\ell}\})$ is a Vinberg pair and let ${\varphi}_{1},\dots ,{\varphi}_{\ell}\in {\mathbb{V}}^{*}$ such that ${M}_{i}={\varphi}_{i}^{-1}\left(0\right)$ and ${\varphi}_{i}\left(C\right)\subset {\mathbb{R}}^{+}$ for $i=1,\dots ,\ell \text{.}$ Hence $C=\{x\in \mathbb{V}\hspace{0.17em}|\hspace{0.17em}{\varphi}_{i}\left(x\right)>0,i=1,\dots ,\ell \}\text{.}$ Let $J\subset \{1,\dots ,\ell \}$ be of facettype by a facet $F\text{.}$ Set ${C}_{J}=\{x\in \mathbb{V}\hspace{0.17em}|\hspace{0.17em}{\varphi}_{i}\left(x\right)>0,j\in J\}\text{.}$ Then:
(i) | $({C}_{J},\left\{{\sigma}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in J\})$ is also a Vinberg pair. |
(ii) | ${C}_{J}=C+{L}_{F}$ (Hence also ${\stackrel{\u203e}{C}}_{J}=\stackrel{\u203e}{C}+{L}_{F}\text{).}$ |
(iii) | ${I}_{J}=I+{L}_{F}\text{.}$ |
(iv) | If ${W}_{J}$ is finite, then $\text{St}\left(J\right)$ is open in $\mathbb{V},$ hence contained in $\stackrel{\circ}{I}\text{.}$ |
(v) | $\text{St}\left(J\right)\cap \stackrel{\circ}{I}$ is open. |
Proof. | |
(i) For $j\in J$ we have ${M}_{j}\in {\mathcal{M}}^{C}$ so there exists a point $x\in \stackrel{\u203e}{C}\cap {M}_{j}$ such that $x\notin {M}_{i}$ for $i\ne j\text{.}$ But then also $x\in {\stackrel{\u203e}{C}}_{j}\cap {M}_{j},$ so ${M}_{j}\in {\mathcal{M}}^{{C}_{J}}\text{.}$ So the polyhedral cone property (2.1)(V1) holds also for the subpair $({C}_{J},\left\{{\sigma}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in J\})\text{.}$ To check the Tits property let $y=w\left(x\right)\in {C}_{J}\cap w\left({C}_{J}\right),$ where $w\in {W}_{J}$ and $x\in {C}_{J}\text{.}$ Let $f\in F,$ choose $\epsilon >0$ such that $\epsilon <\frac{{\varphi}_{i}\left(f\right)}{-{\varphi}_{i}\left(x\right)}$ for all $i\notin J$ with ${\varphi}_{i}\left(x\right)<0$ and $\epsilon <\frac{{\varphi}_{i}\left(f\right)}{-{\varphi}_{i}\left(w\left(x\right)\right)}$ for all $i\notin J$ with ${\varphi}_{i}\left(w\left(x\right)\right)<0\text{.}$ Then ${\varphi}_{i}(f+\epsilon x)={\varphi}_{i}\left(f\right)+\epsilon {\varphi}_{i}\left(x\right)>0$ for all $i\notin J$ and ${\varphi}_{j}(f+\epsilon x)=\epsilon {\varphi}_{j}\left(x\right)>0$ for $j\in J,$ so $f+\epsilon x\in C\text{.}$ Similar we have that $w(f+\epsilon x)=f+\epsilon w\left(x\right)\in C,$ so $w(f+\epsilon x)\in C\cap w\left(C\right),$ hence $w={\text{id}}_{\mathbb{V}}\text{.}$ (ii) To prove ${C}_{J}=C+{L}_{F}$ let first $x\in C$ and $y\in {L}_{F},$ then ${\varphi}_{j}(x+y)={\varphi}_{j}\left(x\right)>0$ for $j\in J$ so $C+{L}_{F}\subset {C}_{J}\text{.}$ Conversely, let $x\in {C}_{J}\text{.}$ Choose $\lambda >\underset{i\in J}{\text{max}}(-\frac{{\varphi}_{i}\left(x\right)}{{\varphi}_{i}\left(f\right)}),$ then ${\varphi}_{i}(x+\lambda f)={\varphi}_{j}\left(x\right)+\lambda {\varphi}_{i}\left(f\right)>0$ if $i\notin J$ and ${\varphi}_{j}(x+\lambda f)={\varphi}_{j}\left(x\right)>0$ if $j\in J,$ so $x+\lambda f\in C,$ hence $x\in C+{L}_{F}\text{.}$ (iii) ${I}_{J}=\bigcup _{w\in {W}_{J}}w\left({\stackrel{\u203e}{C}}_{J}\right)\subset \bigcup _{w\in {W}_{J}}(w\left(\stackrel{\u203e}{C}\right)+{L}_{F})\subset I+{L}_{F}\text{.}$ For the other inclusion it suffices to prove $I\subset {I}_{J}$ i.e. $w\left(\stackrel{\u203e}{C}\right)\subset {I}_{J}$ if $w\in W\text{.}$ Then $w\left(C\right)\subset \stackrel{\circ}{I}$ even $w\left(C\right)\in K(\mathcal{M},\stackrel{\circ}{I})\text{.}$ Let $C\prime $ (respectively ${C}^{\prime \prime}\text{)}$ be the chamber in $K({\mathcal{M}}_{J},\stackrel{\circ}{I})$ that contain $C$ (resp. $w\left(C\right)\text{).}$ Let ${M}_{1},\dots ,{M}_{k}$ be a sequence of elements of ${\mathcal{M}}_{J}$ determining a direct Tits gallery from $C\prime $ to ${C}^{\prime \prime}$ in the hyperplane configuration $(V,\stackrel{\circ}{I},{\mathcal{M}}_{J})\text{.}$ This determines an element $w\prime \in {W}_{J}$ such that ${\left(w\prime \right)}^{-1}\left(w\left(C\right)\right)\subset C\prime \text{.}$ Now $C\prime \subset {D}_{M}\left(C\prime \right)$ for all $M\in {\mathcal{M}}_{J},$ in particular $C\prime \subset {C}_{J}\text{.}$ So ${\left(w\prime \right)}^{-1}\left(w\left(C\right)\right)\subset {C}_{J},$ hence $w\left(C\right)\subset w\prime \left({C}_{J}\right),$ so $w\left(\stackrel{\u203e}{C}\right)\subset w\prime \left({\stackrel{\u203e}{C}}_{J}\right)\subset {I}_{J}\text{.}$ $$\begin{array}{c}\n\n\n\n\n\n\n\n\n\nC\nC\u2032\nC\u2032\u2032\nw(C)\n\n\\ \text{Figure 15.}\end{array}$$(iv) Set $A\u2254\{x\in V\hspace{0.17em}|\hspace{0.17em}{\varphi}_{i}\left(w\left(x\right)\right)>0\hspace{0.17em}\text{for all}\hspace{0.17em}i\notin J\hspace{0.17em}\text{and}\hspace{0.17em}w\in {W}_{J}\}\text{.}$ Then always $\text{St}\left(J\right)\subset A$ and we prove the converse in the case that ${W}_{J}$ is finite (which proves (iv), because if ${W}_{J}$ is finite then $A$ is open in $\mathbb{V}\text{).}$ Let $x\in A\text{.}$ Since ${W}_{J}$ if finite, we have $x\in \mathbb{V}={I}_{J}=\bigcup _{w\in {W}_{J}}w\left({\stackrel{\u203e}{C}}_{J}\right),$ so $x=w\left(y\right)$ for an $y\in {\stackrel{\u203e}{C}}_{J}=\stackrel{\u203e}{C}+{L}_{F}\text{.}$ Let $y=y\prime +{y}^{\prime \prime},$ $y\prime \in \stackrel{\u203e}{C}$ and ${y}^{\prime \prime}\in {L}_{F}\text{.}$ Then $$\begin{array}{ccc}{\varphi}_{i}\left(y\right)& =& {\varphi}_{i}\left({w}^{-1}\left(x\right)\right)>0\phantom{\rule{2em}{0ex}}\text{if}\hspace{0.17em}i\notin J\hspace{0.17em}\text{for}\hspace{0.17em}x\in A,\\ {\varphi}_{j}\left(y\right)& =& {\varphi}_{j}\left(y\prime \right)+{\varphi}_{j}\left({y}^{\prime \prime}\right)={\varphi}_{j}\left(y\prime \right)\ge 0\phantom{\rule{2em}{0ex}}\text{if}\hspace{0.17em}j\in J\text{.}\end{array}$$So $y\in {C}^{J},$ hence $x\in \text{St}\left(J\right)\text{.}$ (v) Let $x\in \text{St}\left(J\right)\cap \stackrel{\circ}{I}\text{.}$ We can suppose that $x\in {C}^{J}\text{.}$ Since $x\in \stackrel{\circ}{I},$ we have $x\in {C}^{J\prime}$ with ${W}_{J\prime}$ finite and $J\prime \subset J\text{.}$ Then $\text{St}\left(J\prime \right)$ is an open neighbourhood of $x$ in $\text{St}\left(J\right)\cap \stackrel{\circ}{I}\text{.}$ $\square $ |
(4.7) Remark. If ${W}_{J}$ is finite, then $J$ is of facettype. (cf. Vinberg [Vin1971] theorem 7. pg. 1114). Hence (v) is true for general $J\text{.}$ As we saw (remark (4.3)) this is also true for (i). Perhaps for (ii) and (iii) too, where ${L}_{F}$ has to be replaced by $\bigcap _{j\in J}{M}_{j}\text{.}$
(4.8) Definition. We have already defined (4.2) the space ${Y}_{J}\u2254Y(\mathbb{V},{\stackrel{\circ}{I}}_{J},{\mathcal{M}}_{F})\text{.}$ We also define: $Y\left(J\right)\u2254Y(\mathbb{V},\stackrel{\circ}{I}\cap \text{St}\left(J\right),\mathcal{M})$ for $J\subset \{1,\dots ,\ell \}\text{.}$ Note that $Y\left(J\right)\subset Y\subset {Y}_{J}$ and $Y\left(J\right)={\text{Im}}^{-1}\left(\text{St}\left(J\right)\right),$ where $\text{Im}:Y\to H\text{.}$
(4.9) Proposition. Let $J\subset \{1,\dots ,\ell \}$ be of facettype. Then there exists a map $r:{Y}_{J}\to Y\left(J\right)$ with properties:
(i) | The map $r$ is a homotopy inverse of the inlusion $i:Y\left(J\right)\subset {Y}_{J}\text{.}$ |
(ii) | For another $J\prime \subset \{1,\dots ,\ell \}$ we have $r\left(Y\left(J\prime \right)\right)\subset Y\left(J\prime \right)\text{.}$ |
Proof. | |
Let $f\in F\text{.}$ Then ${\varphi}_{j}\left(f\right)>0$ for $j\notin J\text{.}$ Choose a closed ball $B$ (in some metric on $\mathbb{V}\text{),}$ such that ${\varphi}_{j}\left(B\right)\subset {\mathbb{R}}^{+}$ for $j\notin J\text{.}$ Then we have $B\cap {\stackrel{\u203e}{C}}_{J}\subset {C}^{J}\text{.}$ For each $y\in {\stackrel{\u203e}{C}}_{J}$ there is a unique $r\left(y\right)\in B\cap {\stackrel{\u203e}{C}}_{J}$ such that $[f,r\left(y\right)]=B\cap [f,y]\text{.}$ Then $r$ is a continuous map ${\stackrel{\u203e}{C}}_{J}\to {C}^{J}\text{.}$ Extend to a map $r:{I}_{J}\to \text{St}\left(J\right)$ by $r\left(y\right)\u2254{w}^{-1}\left(r\left(w\left(y\right)\right)\right),$ where $w\in {W}_{J}$ is such that $w\left(y\right)\in {\stackrel{\u203e}{C}}_{J}\text{.}$ This is well defined by (4.1). We show that $r$ is continuous on ${\stackrel{\circ}{I}}_{J}\text{.}$ Let $H\u2254w{W}_{J\prime}$ be a left coset, where $J\prime \subset J,$ such that ${W}_{J\prime}$ is finite. Now $r$ is continuous on ${\stackrel{\u203e}{C}}_{J}$ and since ${C}^{J\prime}\subset {C}^{J}\subset {\stackrel{\u203e}{C}}_{J},$ it is so on $w.w\prime \left({C}^{J\prime}\right),$ where $w\prime \in {W}_{J\prime}\text{.}$ $\text{Star}\left(H\right)$ is a finite union of these, in $\text{Star}\left(H\right)$ closed sets, so $r$ is continuous on $\text{Star}\left(H\right)\text{.}$ Sets of this type $\text{Star}\left(H\right)$ are open by (4.6)(iv) and cover ${\stackrel{\circ}{I}}_{J},$ hence $r$ is continuous on ${\stackrel{\circ}{I}}_{J}\text{.}$ Since $f\in M$ for each $M\in {\mathcal{M}}_{J}$ and $r\left(y\right)\ne f$ iff $y\ne f,$ we have for $y\in {I}_{J},$ $y\ne f$ and $M\in {\mathcal{M}}_{J}:$ $$\begin{array}{cc}\text{(\$)}& y\in M\iff [y,r\left(y\right)]\cap M\ne \varnothing \iff [y,r\left(y\right)]\subset M\iff r\left(y\right)\in M\text{.}\end{array}$$From ($) it folows that $y$ and $r\left(y\right)$ have the same stabiliser in ${W}_{J}\text{.}$ Since $r\left(y\right)\in \text{St}\left(J\right),$ the stabiliser of $r\left(y\right)$ in $W$ and ${W}_{J}$ is also the same. It follows with the help of (II.2.2)(b) that $r\left({\stackrel{\circ}{I}}_{J}\right)\subset {\stackrel{\circ}{I}}_{J}\text{.}$ Since $r\left({I}_{J}\right)\subset \text{St}\left(J\right)\subset I,$ He have: $$\begin{array}{cc}\text{(\$\$)}& r\left({\stackrel{\circ}{I}}_{J}\right)\subset \text{St}\left(J\right)\cap \stackrel{\circ}{I}\text{.}\end{array}$$By ($) and ($$) we see that we can extend $r$ to a map $r:{Y}_{J}\to Y\left(J\right)$ by $r(x+\sqrt{-1}y)\u2254x+\sqrt{-1}r\left(y\right)\text{.}$ From ($) it follows also that $[z,r\left(z\right)]$ is contained in ${W}_{J}$ (respectively $Y\left(J\right)\text{)}$ if $z\in {Y}_{J}$ (respectively $z\in Y\left(J\right)\text{).}$ From this we see that $i\circ r\simeq {\text{id}}_{{Y}_{J}}$ (respectively $r\circ i\simeq {\text{id}}_{Y\left(J\right)}\text{).}$ Hence $r$ is an homotopy inverse of $i\text{.}$ This proves (i). To prove (ii) it suffices to see that, if $y\in \text{St}\left(J\prime \right),$ then $r\left(y\right)\in \text{St}\left(J\prime \right)\text{.}$ Now $f\in \stackrel{\u203e}{C},$ so $f\in \stackrel{\u203e}{\text{St}\left(J\prime \right)}$ and since $\text{St}\left(J\prime \right)$ is convex we ahve $(f,y]\subset \text{St}\left(J\prime \right)\text{.}$ $\square $ |
Now we come to the study of the abstract Artin groups. Let $W$ be an (abstract) Coxeter group.
(4.10) Definition. The kernel of the projection of the associated Artin group on the Coxeter group: ${A}_{W}\to W$ is called the coloured Artin group and denoted by $C{A}_{W}\text{.}$
We have an exact sequence $1\to C{A}_{W}\to {A}_{W}\to W\to 1\text{.}$ Note that $C{A}_{W}=\pi (Y,\hat{*})$ in the case of a linear Coxeter group, where $\hat{*}\u2254\sqrt{-1}c$ serves as a base point of $Y$ as before.
(4.11) Lemma. Let $J\subset \{1,\dots ,\ell \}\text{.}$ Then there is a natural injective homomorphism $\text{Cf}:C{A}_{{W}_{J}}\to C{A}_{W}\text{.}$
Proof. | |
We may suppose that $W$ is a Tits linear Coxeter group (II.2.11). $\square $ |
Then $J$ is of facettype, hence by lemma (4.9) $Y\left(J\right)\subset {Y}_{J}$ is a homotopy equivalence, so $Y\left(J\right)\subset Y$ has a homotopy left inverse. Hence $\pi (Y\left(J\right),\hat{*})\to \pi (Y,\hat{*})$ is injective. Since $\pi (Y\left(J\right),\hat{*})$ is natural isomorphic to $\pi ({Y}_{J},\hat{*})=C{A}_{{W}_{J}}$ by the homotopy equivalence $Y\left(J\right)\subset {Y}_{J},$ we get the desired injective homomorphism $\text{Cf}:C{A}_{{W}_{J}}\to {A}_{{W}_{J}}\text{.}$
So from now on we can consider $C{A}_{{W}_{J}}$ as a subgroup of $C{A}_{W}\text{.}$
(4.12) Lemma. Let ${J}^{1},{J}^{2}\subset \{1,\dots ,\ell \},$ then $C{A}_{{W}_{{J}^{1}}}\cap C{A}_{{W}_{{J}^{2}}}=C{A}_{{W}_{{J}^{1}\cap {J}^{2}}}$
Proof. | |
We can again suppose that $W$ is a Tits linear group. Consider $\text{Im}:Y\to \stackrel{\circ}{I}\text{.}$ By proposition (4.5)(ii) we have $Y\left({J}^{1}\right)\cap Y\left({J}^{2}\right)={\text{Im}}^{-1}(\text{St}\left({J}^{1}\right)\cap \stackrel{\circ}{I})\cap {\text{Im}}^{-1}(\text{St}\left({J}^{2}\right)\cap \stackrel{\circ}{I})={\text{Im}}^{-1}(\text{St}\left({J}^{1}\right)\cap \text{St}\left({J}^{2}\right)\cap \stackrel{\circ}{I})={\text{Im}}^{-1}(\text{St}({J}^{1}\cap {J}^{2})\cap \stackrel{\circ}{I})\u2255Y({J}^{1}\cap {J}^{2})\text{.}$ Define the abbreviations: $$\begin{array}{ccc}{G}_{12}& \u2254& \pi (Y\left({J}^{1}\right)\cap Y\left({J}^{2}\right),\hat{*})=\pi (Y({J}^{1}\cap {J}^{2}),\hat{*})\cong C{A}_{{W}_{{J}^{1}\cap {J}^{2}}},\\ {G}_{i}& \u2254& \pi (Y\left({J}^{i}\right),\hat{*})=C{A}_{{W}_{{J}^{i}}},\\ G& \u2254& (Y,\hat{*})=C{A}_{W}\end{array}$$Let $r$ be the map ${Y}_{{J}^{1}}\to Y\left({J}^{1}\right)$ as exists by (4.9). Now from (4.9)(ii) it follows that $r\left(Y\left({J}^{2}\right)\right)\subset Y\left({J}^{2}\right)\cap Y\left({J}^{1}\right)=Y({J}^{1}\cap {J}^{2})\text{.}$ So ${r}_{*}\left({G}_{2}\right)\subset {G}_{12}$ and since by (4.9)(i) ${r}_{*}{|}_{{G}_{1}}={\text{id}}_{{G}_{1}},$ we see that ${G}_{1}\cap {G}_{2}={r}_{*}({G}_{1}\cap {G}_{2})\subset {G}_{12}\text{.}$ Since the other inclusion is obvious, we have that ${G}_{1}\cap {G}_{2}={G}_{12},$ what was to be proved. $$\n\n\n\n\n\nG1\n\u222a\n\nG12\u2282\nG1\u2229G2\n\u2282G\n\n\u2229\nG2\n\n\n\n\n\n\n\n\nr*\n\n\n$$$\square $ |
(4.13) Theorem. Let $({A}_{W},\{{s}_{1},\dots ,{s}_{\ell}\})$ be an Artin System.
(i) | If $J\subset \{1,\dots ,\ell \},$ then the subgroup generated by $\left\{{s}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in J\}$ forms with these generators itself an Artin system. In particular the Artin group ${A}_{{W}_{J}}$ can we seen as subgroup of ${A}_{W}\text{.}$ |
(ii) | Let ${J}^{1},{J}^{2}\subset \{1,\dots ,\ell \}\text{.}$ Then ${A}_{{W}_{{J}^{1}}}\cap {A}_{{W}_{{J}^{2}}}={A}_{{W}_{{J}^{1}\cap {J}^{2}}}\text{.}$ |
Proof. | |
Let $f:{A}_{{W}_{J}}\to {A}_{W}$ and $\stackrel{\u203e}{f}:{W}_{J}\to W$ be the natural homomorphisms. We know already that $\stackrel{\u203e}{f}$ is injective. We have a commutative diaqram with exact rows and columns. That $f$ is injective follows from the fact that $\stackrel{\u203e}{f}$ and Cf (4.11) are so and by diagram chasing. This proves the first part or the theorem. $$\begin{array}{ccccccc}& & 1& & & & 1\\ & & \downarrow & & & & \downarrow \\ 1& \u27f6& C{A}_{{W}_{J}}& \u27f6& {A}_{{W}_{J}}& \u27f6& {W}_{J}& \u27f6& 1\\ & & Cf\downarrow & & f\downarrow & & \stackrel{\u203e}{f}\downarrow \\ 1& \u27f6& C{A}_{W}& \u27f6& {A}_{W}& \u27f6& W& \u27f6& 1\end{array}$$(ii) It is clear that ${A}_{{W}_{{J}^{1}}}\cap {A}_{{W}_{{J}^{2}}}\supset {A}_{{W}_{{J}^{1}\cap {J}^{2}}}\text{.}$ To prove the other inclusion suppose that $x\in {A}_{{W}_{{J}^{1}}}\cap {A}_{{W}_{{J}^{2}}}\text{.}$ Then $\pi \left(x\right)\u2255w\in {W}_{{J}^{1}}\cap {W}_{{J}^{2}}={W}_{{J}^{1}\cap {J}^{2}}\text{.}$ By remark (3.19) we can consider $w$ as an element of ${A}_{{W}_{{J}^{1}\cap {J}^{2}}}\text{.}$ Then $x{w}^{-1}\in C{A}_{{W}_{{J}^{1}}}\cap C{A}_{{W}_{{J}^{2}}}=C{A}_{{W}_{{J}^{1}\cap {J}^{2}}}\subset {A}_{{W}_{{J}^{1}\cap {J}^{2}}}\text{.}$ Since $w\in {A}_{{W}_{{J}^{1}\cap {J}^{2}}}$ we also have $x\in {A}_{{W}_{{J}^{1}\cap {J}^{2}}}\text{.}$ $\square $ |
This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.