Discrete complex reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 12 May 2014

Notes and References

This is an excerpt of the lecture notes Discrete complex reflection groups by V.L. Popov. Lectures delivered at the Mathematical Institute, Rijksuniversiteit Utrecht, October 1980.

Notation and formulation of the problem

We assume in this chapter that the ground field $k$ be $ℝ$ or $ℂ\text{.}$

Notation

Let $E$ be an affine space over $k,$ $\text{dim}\hspace{0.17em}E=n,$ and $V$ be its space of translations. If $v\in V,$ we denote by ${\gamma }_v$ the corresponding translation of $E,$ i.e. $${\gamma }_v\left(a\right)=a+v,a\in E\text{.}$$ Let $A\left(E\right)$ be the group of all affine transformations of $E$ and $$\text{Tran}\hspace{0.17em}A\left(E\right)=\left\{{\gamma }_v\hspace{0.17em}\right|\hspace{0.17em}v\in V\}\text{.}$$ We denote by $GL\left(V\right)·V$ the natural semidirect product of $GL\left(V\right)$ and $V\text{.}$ Its elements are pairs $(P,v),$ where $P\in GL\left(V\right),$ $v\in V,$ and the group operations are given by formulas $$\begin{array}{ccc}(P,v)(Q,w)& =& (PQ,Pw+v),\\ {(P,v)}^{-1}& =& (P^{-1},-P^{-1}v)\text{.}\end{array}$$ Let $$\text{Lin}:\hspace{0.17em}A\left(E\right)\to GL\left(V\right)$$ be the standard homomorphism defined by formula $$\gamma (a+v)=\gamma \left(a\right)+\left(\text{Lin}\hspace{0.17em}\gamma \right)v,\gamma \in A\left(E\right),a\in E,v\in V\text{.}$$ If we take a point $a\in E$ as origin, we obtain an isomorphism $${\kappa }_a:\hspace{0.17em}A\left(E\right)\to GL\left(V\right)·V$$ given by the formula $${\kappa }_a\left(\gamma \right)=(\text{Lin}\hspace{0.17em}\gamma ,\gamma \left(a\right)-a)\text{.}$$ Identifying $A\left(E\right)$ and $GL\left(V\right)·V$ by means of ${\kappa }_a,$ we obtain the action of $GL\left(V\right)·V$ on $E$ given by the formula $$(P,v)q=a+P(q-a)+v,q\in E\text{.}$$ The dependence on $a$ is given by $${\kappa }_b\left(\gamma \right)={\kappa }_a\left({\gamma }_{a-b}\gamma {\gamma }_{b-a}\right),\gamma \in A\left(E\right),b\in E\text{.}$$ For every $\gamma \in A\left(E\right)$ and $P\in GL\left(V\right)$ we use the notations $$\begin{array}{ccc}{H}_{\gamma }& =& \{a\in E\hspace{0.17em}|\hspace{0.17em}\gamma \left(a\right)=a\},\\ H_P& =& \{v\in V\hspace{0.17em}|\hspace{0.17em}Pv=v\}\text{.}\end{array}$$ These are subspaces of $E$ and $V$ respectively.

Let $⟨\hspace{0.17em}|\hspace{0.17em}⟩$ be a positive definite inner product on $V,$ i.e. $V$ is an euclidian $\text{(}k=ℝ\text{),}$ resp. hermitian $\text{(}k=ℂ\text{)}$ linear space with respect to $⟨\hspace{0.17em}|\hspace{0.17em}⟩$ (linear in the first coordinate). Let also $U\left(V\right)=\{P\in GL\left(V\right)\hspace{0.17em}|\hspace{0.17em}P\hspace{0.17em}\text{preserves}\hspace{0.17em}⟨\hspace{0.17em}|\hspace{0.17em}⟩\}\text{;}$ this is a compact group. The space $E$ becomes a euclidean, resp. hermitian affine metric space with respect to the distance given by the formula $$\rho (a,b)=\sqrt{⟨a-b\hspace{0.17em}|\hspace{0.17em}a-b⟩},a,b\in E\text{.}$$

Motions and reflections

We say that $\gamma \in A\left(E\right)$ is a motion of $E$ if $\gamma$ preserves the distance $\rho \text{.}$ It is easy to see that $\gamma$ is a motion iff $\text{Lin}\hspace{0.17em}\gamma \in U\left(V\right)\text{.}$

Definition. An affine reflection $\gamma \in A\left(E\right)$ is an element with the properties:

1)	$\gamma$ is a motion,
2)	$\gamma$ has finite order,
3)	$\text{codim}\hspace{0.17em}{H}_{\gamma }=1\text{.}$

A linear reflection $R\in GL\left(V\right)$ is an element with the properties:

1)	$R\in U\left(V\right),$
2)	$R$ has finite order,
3)	$\text{codim}\hspace{0.17em}{H}_R=1\text{.}$

The subspaces $H_{\gamma }$ and $H_R$ are called the mirrors of $\gamma ,$ resp. $R\text{.}$

Sometimes we s!1all simply say reflection when it is clear what we are talking about.

If $R$ is a linear reflection then the line $${\ell }_R=\{v\in V\hspace{0.17em}|\hspace{0.17em}v\perp H_R\}$$ is called the root line of $R\text{.}$ If $v\in {\ell }_R,$ $⟨v\hspace{0.17em}|\hspace{0.17em}v⟩=1,$ then $Rv=\theta v,$ where $0\ne 1$ is a primitive root of $1$ (if $k=ℝ$ then $\theta =-1,$ if $k=ℂ$ then $\theta$ may be arbitrary). The pair $(v,\theta )$ determines $R$ completely and every pair $(u,\eta ),$ with $⟨u\hspace{0.17em}|\hspace{0.17em}u⟩=1,$ $\eta \ne 1$ a primitive root of unity $\text{(}=-1$ if $k=ℝ\text{),}$ can be obtained in such a way from a reflection. We shall write $$R=R_{v,\theta }$$ Some properties of the reflections are contained in the following

Proposition. Let $\gamma \in A\left(E\right),$ $a\in E$ and ${\kappa }_a\left(\gamma \right)=(R,v)\text{.}$ Then

1)	γ is a reflection iff $R$ is a reflection and $v\perp H_R\text{.}$
2)	If $\gamma$ is a reflection and $R=R_{e,\theta }$ then $$H_{\gamma }=a+H_R+{(1-\theta )}^{-1}v\text{.}$$
3)	$R_{e,\theta }v=v-(1-\theta )⟨v\hspace{0.17em}|\hspace{0.17em}e⟩e\text{.}$
4)	If $\gamma$ is a reflection and $\delta$ is a motion then $\delta \gamma {\delta }^{-1}$ is a reflection.

		Proof.
	Proof is left to the reader. $\square$

Main problem

We shall say that a subgroup $W$ of $A\left(E\right)$ is an $r\text{-group}$ if it is discrete and generated by affine reflections.

If $E$ and $E\prime$ are two affine spaces and $G\subseteq A\left(E\right)$ and $G\prime \subseteq A\left(E\prime \right)$ are two arbitrary subgroups, then we shall say that $G$ and $G\prime$ are equivalent if there exists an affine bijection $\varphi :E\to E\prime$ such that $$W\prime =\varphi W{\varphi }^{-1}\text{.}$$

This means that after identifying $E$ and $E\prime$ by means of an arbitrary fixed isomorphism, the groups $G$ and $G\prime ,$ as subgroups of $A\left(E\right),$ have to be conjugate in $A\left(E\right)\text{.}$

We want to emphasize here that even when $E$ and $E\prime$ are affine metric spaces, $\varphi$ need not to be distance preserving.

Our main concern in these lectures will be to classify $r\text{-groups}$ up to equivalence.

We shall show now that in solving this problem one can restrict attention to irreducible groups.

Irreducibility

Let $W$ be a subgroup of $A\left(E\right)\text{.}$ We shall say that $W$ is reducible if there exist affine metric spaces $E_j,$ $j=1,\dots ,m,m\ge 2,$ and subgroups $W_j$ of $A\left(E_j\right)$ such that $W$ is equivalent to $W_1\times \cdots \times W_m\subseteq A(E_1\times \cdots \times E_m)\text{.}$ Otherwise $W$ is called irreducible. Clearly, every group is equivalent to a product of irreducible groups (but its decomposition need not be unique).

Theorem. Let $W\subseteq A\left(E\right)$ be a nontrivial subgroup generated by affine reflections (possibly not discrete). Then:

a)	$W$ is equivalent to a product $W_1\times \cdots \times W_m$ where $W_j,$ $1\le j\le m,$ are irreducible groups, and $W_j$ is either generated by affine reflections or trivial (hence $1\text{-dimensional),}$ but not all $W_j$ are trivial.
b)	$W$ is irreducible iff $\text{Lin}\hspace{0.17em}W$ is an irreducible linear group (generated by linear reflections).
c)	$W_j,$ $1\le j\le m,$ are uniquely defined up to equivalence and order.
d)	Every product of the type described in a) is a group generated by reflections.

		Proof.
	a) The statement follows from the equality $$H_{({\gamma }_1,\dots ,{\gamma }_m)}=H_{{\gamma }_1}\times \cdots \times H_{{\gamma }_m}$$ (hence $({\gamma }_1,\dots ,{\gamma }_m)$ is a reflection iff one and only one of ${\gamma }_j,$ $1\le j\le m,$ is a reflection and the others are 1) b) The "if" part is obvious. Let us prove the "only if" part. As the group $W$ is generated by reflections, the group $\text{Lin}\hspace{0.17em}W$ lies in $U\left(V\right)\text{.}$ Therefore, $\text{Lin}\hspace{0.17em}W$ is a completely reducible linear group. Let $$V=\underset{j=1}{\overset{m}{\oplus }}{V}_j$$ where $V_j,$ $1\le j\le m,$ are irreducible $\text{Lin}\hspace{0.17em}W\text{-modules.}$ Consider the subspaces $$E_j=a+V_j,1\le j\le m,$$ where $a\in E$ is an origin, and let $${\pi }_j:\hspace{0.17em}W\to A\left(E_j\right),1\le j\le m,$$ be the morphism given by the formula $${\pi }_j\left(\gamma \right)={\kappa }_a^{-1}(\text{Lin}\hspace{0.17em}\gamma {|}_{V_j},p_j(\gamma \left(a\right)-a))$$ (here $p_j:V\to V_j$ is the natural projection). Let $W_j={\pi }_j\left(W\right)\text{.}$ Then it is not difficult to check that the map $$\varphi :\hspace{0.17em}E\to E_1\times \cdots \times E_m,$$ given by the formula $$\varphi \left(q\right)=(a+p_1(q-a),\dots ,a+p_m(q-a),q\in E,)$$ defines an equivalence of $W$ and $W_1\times \cdots \times W_m\text{.}$ c) Suppose that $W\subset A\left(E\right)$ and $W\prime \subset A\left(E\prime \right)$ are two equivalent groups generated by affine reflections and let $\varphi :E\to E\prime$ establish the equivalence of these groups. Let $W=W_1\times \cdots \times W_r$ and $W\prime =W_1^{\prime }\times \cdots \times W_s^{\prime }$ be decompositions into products of irreducible groups and let $E=E_1\times \cdots \times E_r,$ $E\prime =E_1^{\prime }\times \cdots \times E_s^{\prime },$ $V=V_1\oplus \cdots \oplus V_r,$ $V\prime =V_1^{\prime }\oplus \cdots \oplus V_s^{\prime },$ be the corresponding decompositions of the affine spaces and its spaces of translations. We consider $W_j$ and $W_1^{\prime }$ as subgroups of $W$ and $W\prime$ resp. It is clear that $\text{Lin}\hspace{0.17em}\psi$ yields the equivalence of the linear groups $\text{Lin}\hspace{0.17em}W$ and $\text{Lin}\hspace{0.17em}W\prime$ (in the usual sense). Hence $\left(\text{Lin}\hspace{0.17em}\psi \right)V_j$ is a simple $\text{Lin}\hspace{0.17em}W\prime \text{-submodule}$ of $V\prime$ for every $j\text{.}$ Let $\gamma \in W$ be a reflection. Then $\gamma \in W_p$ for some $p,$ see a) above. But $\psi \gamma {\psi }^{-1}=\gamma \prime$ is also a reflection (inside $W\prime \text{).}$ Therefore $\gamma \prime \in W_q^{\prime }$ for some $q\text{.}$ The root line of $\text{Lin}\hspace{0.17em}\gamma \prime$ is contained in $V_q^{\prime }$ and this line is $\left(\text{Lin}\hspace{0.17em}\psi \right)\ell ,$ where $\ell$ is a root line of $\text{Lin}\hspace{0.17em}\gamma \text{.}$ Clearly, $\ell \subset V_p\text{.}$ Hence $$V_q^{\prime }\cap \left(\text{Lin}\hspace{0.17em}\psi \right)V_p\ne 0\text{.}$$ It follows now from the irreducibility that in fact $$V_q^{\prime }=\left(\text{Lin}\hspace{0.17em}\gamma \right)V_p\text{.}$$ Therefore $$\psi V_p{\psi }^{-1}\subset W_q^{\prime }$$ (because $W_p$ is generated by all reflections $\tau$ for which $\text{Lin}\hspace{0.17em}\tau$ has its root line in $V_p\text{).}$ The same proof is valid for the inverse inclusion. Hence, $$\psi W_p{\psi }^{-1}=W_q^{\prime }$$ and we can proceed by induction. d) This is clear (see the equality in a)). $\square$

Therefore we shall from now on only consider the case of irreducible $r\text{-groups.}$

What is already known: case $k=ℝ\text{.}$

Let $W$ be an irreducible $r\text{-group}$ in $A\left(E\right)\text{.}$

One has two possibilities: either $W$ is finite or $W$ is infinite.

If $W$ is finite then there exists a point in $E$ which is fixed under $W\text{.}$ Indeed, let $a\in E$ be an arbitrary point. Then $$b=a+\frac{1}{\left|W\right|}\sum _{\gamma \in W}(\gamma \left(a\right)-a)$$ is fixed under $W\text{.}$ Therefore ${\kappa }_b$ gives an isomorphism of $W$ with $\text{Lin}\hspace{0.17em}W,$ see Section (1.1), i.e. $W$ is a linear group generated by reflections (we identify $E$ and $V$ by choosing the origin b).

We shall consider now the case $k=ℝ\text{.}$

A beautiful classical theory concerning this case was developed by Coxeter, Witt, Stiefel, see [CRe1961], the results of which we shall recapitulate here.

1) $W$ is finite.

$W$ is either a Weyl group of an irreducible root system, or a dihedral group, or one of two exceptional groups $H_3$ or $H_4\text{.}$

The description of these groups is usually given by means of their Coxeter graphs. This is done in the following way. It is known that $W$ is generated by $n$ elements $R_j,$ $1\le j\le n,$ which are reflections in the faces of a Weyl chamber $C\text{.}$ There exists a unique set of vectors $e_j,$ $1\le j\le n,$ of unit length with the property: $R_j=R_{e_j,{\theta }_j}$ (where in fact ${\theta }_j=-1\text{)}$ and $C=\{v\in V\hspace{0.17em}|\hspace{0.17em}⟨e_j\hspace{0.17em}|\hspace{0.17em}v⟩>0,1\le j\le n\}\text{.}$ The angle between the mirrors $H_{R_i}$ and $H_{R_j}$ is of the form $\frac{\pi }{m_{ij}},$ $m_{ij}\in \mathbb{Z}\text{;}$ $m_{ij}\ge 2\text{.}$ The nodes of the Coxeter graph of $W$ are in bijective correspondence with the reflections $R_j,$ $1\le j\le n\text{.}$ Two nodes $R_i$ and $R_j$ are connected by an edge iff $m_{ij}\ge 3\text{.}$ The weight of this edge is equal to $m_{ij}$ (if $m_{ij}=3$ then the weight is usually omitted). The complete list of finite irreducible $r\text{-groups}$ is given by the following table of Coxeter graphs. $$\begin{array}{cccc}\begin{array}{c} \end{array}& A_n,& \begin{array}{c} 4 \end{array}& F_4,\\ \begin{array}{c} 4 \end{array}& B_n=C_n,& \begin{array}{c} 6 \end{array}& G_2,\\ \begin{array}{c} \end{array}& D_n,& \begin{array}{c} 5 \end{array}& H_3,\\ \begin{array}{c} \end{array}& E_6,& \begin{array}{c} 5 \end{array}& H_4,\\ \begin{array}{c} \end{array}& E_7,& \begin{array}{c} p \end{array}& I_2\left(p\right),\hspace{0.17em}p=5\hspace{0.17em}\text{or}\hspace{0.17em}p\ge 7\text{.}\\ \begin{array}{c} \end{array}& E_8,\end{array}$$ Let us consider the numbers $$c_{ij}=(1-{\theta }_i)(1-{\theta }_j)⟨e_i\hspace{0.17em}|\hspace{0.17em}{e}_j⟩⟨e_j\hspace{0.17em}|\hspace{0.17em}{e}_i⟩=4{\text{cos}}^2\frac{\pi }{m_{ij}}\text{.}$$ One can change the weight $m_{ij}$ to the numbers $c_{ij}$ for all of the edges of the Coxeter graph. In this manner another weighted graph results. Clearly, one graph determined the other. We shall show later the newly obtained graphs can be generalized to the complex

2) $W$ is infinite.

$W$ is an affine Weyl group of an irreducible root system.

This group is a semidirect product of $\text{Lin}\hspace{0.17em}W,$ which is a (finite) Weyl group of a certain root system $R,$ and the lattice of rank $n$ generated by the dual root system $Ř\text{.}$ The groups $\text{Lin}\hspace{0.17em}W,$ thus obtained are distinguished from the others of the list above in the following way (see [Bou1968], Ch. VI):

Theorem. Let $K\subset GL\left(V\right)$ be an irreducible finite $r\text{-group.}$ Then the following properties are equivalent:

a)	$K=\text{Lin}\hspace{0.17em}W$ where $W$ is an infinite $r\text{-group.}$
b)	There exists a $K\text{-invariant}$ lattice in $V$ of rank $n\text{.}$
c)	$K$ is defined over $\mathbb{Q}\text{.}$
d)	$K$ is the Weyl group of a certain irreducible root system, i.e. a group whose Coxeter graph is one of $A_n,$ $B_n,$ $D_n,$ $E_6,$ $E_7,$ $E_8,$ $F_4,$ $G_2\text{.}$
e)	All the numbers $c_{ij}$ lie in $\mathbb{Z}\text{.}$
f)	The ring with unity, generated over $\mathbb{Z}$ by all of the numbers $c_{ij},$ coincides with $\mathbb{Z}\text{.}$

What is already known: case $k=ℂ\text{.}$

Let $k=ℂ$ and $W\subset A\left(E\right)$ be an irreducible $r\text{-group.}$

1) $W$ is finite.

Shephard and Todd, [STo1954], gave the complete list of such groups. A modern and unified approach was presented by Cohen, [Coh1976].

We shall describe this classification in a form that is more convenient for us, i.e. by means of certain graphs (as it was done in the real case).

Let $R_j=R_{e_j,{\theta }_j},$ $1\le j\le s,$ be a generating system of reflections of $W\text{.}$ We can assume that $${\theta }_j=e^{\frac{2\pi i}{m_j}}\text{.}$$ Therefore this system (and hence $W\text{)}$ is uniquely defined by the system of lines ${\ell }_{R_j}$ in $V$ with the multiplicities $m_j,$ $1\le j\le s\text{.}$

It is well known that an arbitrary set of vectors in $V$ is uniquely (up to isometry) defined by means of a certain set of numbers (more precisely by the corresponding Gram matrix). Let us show that the same is true for an arbitrary set of lines in $V$ (i.e. points of the corresponding projective space).

Proposition. Let ${\left\{{\ell }_j\right\}}_{j\in J}$ and ${\left\{{\ell }_j^{!}\right\}}_{j\in J}$ be two sets of lines in $V$ and let $e_j\in {\ell }_j,$ $e_j^{!}\in {\ell }_j^{!}$ be arbitrary vectors with $1=⟨e_j\hspace{0.17em}|\hspace{0.17em}{e}_j⟩=⟨e_j^{!}\hspace{0.17em}|\hspace{0.17em}{e}_j^{!}⟩\text{.}$ For any finite set of indices $j_1,\dots ,j_m\in J$ let us consider the numbers $$h_{j_1\cdots j_m}=⟨e_{j_1}\hspace{0.17em}|\hspace{0.17em}{e}_{j_2}⟩⟨e_{j_2}\hspace{0.17em}|\hspace{0.17em}{e}_{j_3}⟩\cdots ⟨e_{j_{m-1}}\hspace{0.17em}|\hspace{0.17em}{e}_{j_m}⟩⟨e_{j_m}\hspace{0.17em}|\hspace{0.17em}{e}_{j_1}⟩$$ and $$h_{j_1\cdots j_m}^{!}=⟨e_{j_1}^{!}\hspace{0.17em}|\hspace{0.17em}{e}_{j_2}^{!}⟩⟨e_{j_2}^{!}\hspace{0.17em}|\hspace{0.17em}{e}_{j_3}^{!}⟩\cdots ⟨e_{j_{m-1}}^{!}\hspace{0.17em}|\hspace{0.17em}{e}_{j_m}^{!}⟩⟨e_{j_m}^{!}\hspace{0.17em}|\hspace{0.17em}{e}_{j_1}^{!}⟩\text{.}$$ Then $h_{j_1\cdots j_m},$ resp. $h_{j_1\cdots j_m}^{!},$ is independent of the choice of the vectors $e_j,$ resp. $e_j^{!},$ $j\in J\text{.}$ Moreover, the systems ${\left\{{\ell }_j\right\}}_{j\in J}$ and ${\left\{{\ell }_j^{\prime }\right\}}_{j\in J}$ are isometric (i.e. $g{\ell }_j={\ell }_j^{\prime },$ $j\in J,$ for a certain $g\in U\left(V\right)\text{)}$ if and only if $$\begin{array}{cc}{h}_{j_1\cdots j_m}=h_{j_1\cdots j_m}^{!}& (*)\end{array}$$ for arbitrary $j_1,\dots ,j_m\in J\text{.}$

		Proof.
	We need only prove that if $(*)$ is fulfilled then for every $i\in J$ there exists a number ${\lambda }_i\in ℂ$ such that $$⟨e_j^{!}\hspace{0.17em}|\hspace{0.17em}{e}_k^{!}⟩=⟨{\lambda }_j{e}_j\hspace{0.17em}|\hspace{0.17em}{\lambda }_k{e}_k⟩$$ for every $j,k\in J,$ i.e. the systems ${\left\{e_j^{!}\right\}}_{j\in J}$ and ${\left\{{\lambda }_j{e}_j\right\}}_{j\in J}$ have the same Gram matrix (all the other statements are evident). Let us fix an index $t\in J\text{.}$ It is not difficult to see that one can assume that the system ${\left\{{\ell }_j\right\}}_{j\in J}$ is "connected", i.e. there exists for every $j\in J$ a sequence $j_1,\dots ,j_m\in J$ such that $j_1=j,$ $j_m=t$ and $⟨e_{j_s}\hspace{0.17em}|\hspace{0.17em}{e}_{j_{s+1}}⟩\ne 0,$ $s=1,\dots ,m-1\text{.}$ It is now a matter of straightforward computation to check that one can take $${\lambda }_j=\prod _{s=1}^{m-1}\frac{⟨e_{j_{s+1}}^{!}\hspace{0.17em}|\hspace{0.17em}{e}_{j_s}^{!}⟩}{⟨e_{j_{s+1}}\hspace{0.17em}|\hspace{0.17em}{e}_{j_s}⟩}\text{.}$$ $\square$

We have seen above that an $r\text{-group}$ is defined by a system ${\{{\ell }_j,m_j\}}_{j\in J}$ of lines ${\ell }_j\in V$ with the multiplicities $m_j\in \mathbb{Z}\text{.}$ It follows from the proposition that such a system is uniquely (up to isometry) defined by a system of numbers $$c_{j_1\cdots j_m}=h_{j_1\cdots j_m}\prod _{s=1}^m(1-e^{2\pi i/m_{js}})$$ (one can derive from these numbers the multiplicities because $c_j=1-e^{2\pi i/m_j}\text{).}$ It will become clear later on why $h_{j_1\cdots j_m}$ is multiplied by $\prod _{s=1}^m(1-e^{2\pi i/m_{js}})$ (and not, say by $\prod _{s=1}^m{e}^{2\pi i/m_{js}}\text{);}$ the numbers $c_{j_1\cdots j_m}$ are of great importance in the whole theory. We call them cyclic products.

So the group $W$ (with a fixed generating system of reflections) is uniquely (up to equivalence) defined by the corresponding set of cyclic products. As a matter of fact one only needs to know the so called simple cyclic products $c_{j_1\cdots j_m},$ i.e. those with all indices $j_1,\dots ,j_m$ distinct, because $$c_{l_1\cdots l_{p-1}l{l}_{p+1}\cdots l_{q-1}l{l}_{q+1}\cdots l_r}=c_{l_1\cdots l_{p-1}l{l}_{q+1}\cdots l_r}·c_{l{l}_{p+1}\cdots l_{q-1}}·$$ $$\begin{array}{c} l l1 l2 lp-1 lp-1 lp+1 lq+1 lp \end{array}$$

We want to point out here several properties of the cyclic products.

a) If $j_1^{\prime },\dots ,j_m^{\prime }$ is a cyclic permutation of $j_1,\dots ,j_m$ then $$c_{j_1\cdots j_m}=c_{j_1^{\prime }\cdots j_m^{\prime }}\text{.}$$ In other words, $c_{j_1\cdots j_m}$ depends only on a cycle $\sigma =(j_1,\dots ,j_m)\text{.}$ We use therefore the notation $$c_{\sigma }=c_{j_1\cdots j_m}\text{.}$$ In particular, $$c_{j_k}=c_{k_j}\text{.}$$ b) If $\sigma =(j_1,\dots ,j_m)$ then $$c_{\sigma }{c}_{\sigma -1}=c_{j_1{j}_2}{c}_{j_2{j}_3}\cdots c_{j_{m-1}{j}_m}{c}_{j_m{j}_1}$$ c) One can reconstruct all the simple cyclic products (hence all the cyclic products) only from the "homologically independent" ones. The following formula and drawing may illustrate what we have in mind: $$c_{l_1{l}_2\cdots l_p{j}_s{j}_{s-1}\cdots j_1}·c_{l_p{l}_{p+1}\cdots l_{q-1}{l}_q{l}_1{j}_1{j}_2\cdots j_s}=c_{l_1{l}_2\cdots l_{p-1}{l}_p{l}_{p+1}\cdots l_{q-1}{l}_q}·c_{l_1{j}_1}·c_{j_1{j}_2}\cdots c_{j_{s-1}{j}_s}·c_{j_s{l}_p}$$ $$\begin{array}{c} lq-1 lq l1 l2 l3 j1 j2 js-1 js lp+2 lp+1 lp lp-1 lp-2 \end{array}$$

A system of lines ${\ell }_j\in V$ with the multiplicities $m_j,$ $j\in J,$ can be described by a graph as follows.

The nodes of the graph are in bijective correspondence with the lines ${\ell }_j,$ $j\in J\text{.}$ If a node represents the line ${\ell }_j$ then this node has the weight $m_j\text{.}$ Two nodes ${\ell }_j$ and ${\ell }_k$ are connected by an edge iff $c_{jk}\ne 0$ and if they are connected then the weight of the edge is equal to $c_{jk}\text{.}$ It will be convenient to omit the weight of the node, resp. the edge, if it is equal to $2,$ resp. $1\text{.}$ Moreover, every simple cycle of this graph is supplied with an arbitrary (but fixed) orientation and has weight equal to the corresponding cycle product: $$\begin{array}{c} cj1j2⋯js mj2 mj1 mjs mjk-1 mjk mjk+1 cj1j2 cj1js cjk-1jk cjkjk+1 \end{array}$$

Therefore we now have a way to represent a finite $r\text{-group}$ $W\subset GL\left(V\right)$ with a fixed generating system of reflections by means of a graph corresponding to the system of lines ${\ell }_{R_j}\subset V$ with the multiciplicities $m_j,$ $1\le j\le s\text{.}$ (Of course, using another system of generators one obtains another graph which represents the same group. This nonuniqueness in the representations of the group by means of its graph occurs because, contrary to the real case, there is no known canonical method for constructing a generating system of reflections of a finite complex $r\text{-group.}$ The problem of finding such a method is still unsolved and seems to be very interesting). It is easy to see that, $W$ being irreducible, the graph is connected. This graph is called the graph of the group $W$ (with respect to a fixed generating system of reflections).

The classification of the finite complex irreducible $r\text{-groups}$ $W$ was given in [STo1954], [Coh1976] by means of generating systems of reflections. It is now a matter of more of less straightforward computation to reformulate the result by means of the graphs. We need several notations to formulate the corresponding theorem: $$\begin{array}{c}\text{Notation}\text{:}\\ \omega =e^{2\pi i/3}\\ \eta =e^{2\pi i/5}& & {\zeta }_m=e^{2\pi i/m}\\ \epsilon =e^{2\pi i/8}\end{array}$$

It appears a posteriori that all the graphs under consideration are planar and either have no simple cycles or have only one such cycle (of length 3). We assume that this cycle is counterclockwise oriented.

We also fix a numbering of the nodes of the graph (in an arbitrary fashion). The number of the node is written beside the node (but the weight of the node is written inside).

In the table below the ring with unity generated over $\mathbb{Z}$ by all cyclic products is also given. We need this ring later on; it plays an important role in the theory and does not depend on the choice of the generating system of reflections (and hence on the graph that represents the group).

Theorem (classification of irreducible complex finite $r\text{-groups).}$ Any irreducible complex finite $r\text{-groups}$ (with respect to a fixed generating system of reflections) corresponds to a graph of Table 1 (our numbering of the groups coincides with the one of Shephard and Todd [STo1954]; the notation of types is as in [STo1954], [Coh1976], [Cox1974]):

Table 1
The irreducible complex finite $r\text{-groups.}$
No	Type	Graph	Ring generated by cyclic products	$\text{dim}\hspace{0.17em}V$
1	$A_s,$ $s\ge 1$	$\begin{array}{c} 1 s \end{array}$	$\mathbb{Z}$	$s$
2	$G(m,1,s),$ $m\ge 2,$ $s\ge 2,$ type $B_s=C_s$ if $m=2$	$\begin{array}{c} 1 2 s 1-ζm m \end{array}$ $\begin{array}{c} 1 2 s 2 \end{array}$ if $m=2$	$\mathbb{Z}\left[e^{2\pi i/m}\right],$ $\mathbb{Z}\left[\omega \right]$ if $m=3,6,$ $\mathbb{Z}\left[i\right]$ if $m=4,$ $\mathbb{Z}$ if $m=2$	$s$
2	$G(m,m,s),$ $m\ge 2,$ $s\ge 3,$ type $D_s$ if $m=2$	$\begin{array}{c} 1 2 3 s 4cos2πm 2ζ2mcosπm ζ2m=eπi/m \end{array}$	$\mathbb{Z}\left[e^{2\pi i/m}\right],$ $\mathbb{Z}\left[\omega \right]$ if $m=3,6,$ $\mathbb{Z}\left[i\right]$ if $m=4,$	$s$
		$\begin{array}{c} 1 2 3 s \end{array}$ if $m=2$	$\mathbb{Z}$ if $m=2$
2	$G(m,m,2)=I_2\left(m\right),$ $m\ge 3,$ $A_2$ if $m=3,$ $B_2$ if $m=4,$ $G_2$ if $m=6$	$\begin{array}{c} 1 2 4cos2πm \end{array}$ $\begin{array}{c} 1 2 \end{array}$ if $m=3,$ $\begin{array}{c} 1 2 2 \end{array}$ if $m=4,$ $\begin{array}{c} 1 2 6 \end{array}$ if $m=6$	$\mathbb{Z}\left[4{\text{cos}}^2\frac{\pi }{m}\right],$ $\mathbb{Z}$ if $m=3,4,6$	2
2	$G(m,p,s-1),$ $m\ge 2,$ $s\ge 4,$ $p|m,$ $p\ne 1,m$	$\begin{array}{c} 1 2 3 s-1 s m/p 4cos2πm 2ζ2mcosπm 1-ζmp \end{array}$	$\mathbb{Z}\left[e^{2\pi i/m}\right],$ $\mathbb{Z}\left[i\right]$ if $m=4,$ $p=2,$ $\mathbb{Z}\left[\omega \right]$ if $m=6,$ $p=2$ and $m=6,$ $p=3$	$s-1$
2	$G(m,p,2),$ $m\ge 2,$ $p|m,$ $p\ne 1,m$	$\begin{array}{c} 1 2 3 m/p 4cos2πm 2(1-ζmp)ζ2mcosπm 1-ζmp 1-ζmp \end{array}$	$\mathbb{Z}[{\zeta }_m,2\text{cos}\frac{\pi }{m},{\zeta }_m(1-{\zeta }_m^p)],$ $\mathbb{Z}\left[2i\right]$ if $m=4,$ $p=2,$ $\mathbb{Z}\left[\omega \right]$ if $m=6,$ $p=2,$ $\mathbb{Z}\left[2\omega \right]$ if $m=6,$ $p=3$	$2$
3	${\left[\hspace{0.17em}\right]}^m$	$\begin{array}{c} 1 m \end{array}$	$\mathbb{Z}\left[e^{2\pi i/m}\right],$ $\mathbb{Z}\left[\omega \right]$ if $m=6,3,$ $\mathbb{Z}\left[i\right]$ if $m=4,$ $\mathbb{Z}$ if $m=2$	$1$
4	$3\left[3\right]3$	$\begin{array}{c} 1 2 -ω 3 3 \end{array}$	$\mathbb{Z}\left[\omega \right]$	$2$
5	$3\left[4\right]3$	$\begin{array}{c} -2ω 3 3 \end{array}$	$\mathbb{Z}\left[\omega \right]$	$2$
6	$3\left[6\right]2$	$\begin{array}{c} 1 2 iω2+1-ω 3 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/12}\right]$	$2$
7	${⟨3,3,2⟩}_6$	$\begin{array}{c} 1 2 3 iω2+1-ω -2(i+ω) iω2+1-ω -2ω 3 3 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/12}\right]$	$2$
8	$4\left[3\right]4$	$\begin{array}{c} 1 2 -i 4 4 \end{array}$	$\mathbb{Z}\left[i\right]$	$2$
9	$4\left[6\right]2$	$\begin{array}{c} 1 2 1-i-ε3 4 \end{array}$	$\mathbb{Z}\left[\epsilon \right]$	$2$
10	$4\left[4\right]3$	$\begin{array}{c} 1 2 -i-ω 1 4 3 3 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/12}\right]$	$2$
11	${⟨4,3,2⟩}_{12}$	$\begin{array}{c} 1 2 3 iε-i+1 -2iεω2-ε-i+ 2iω2-ω 2iω2-ω+1 -i-ω 4 3 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/24}\right]$	$2$
12	$GL(2,3)$	$\begin{array}{c} 1 2 3 2 4+i2 3 3 \end{array}$	$\mathbb{Z}\left[i\sqrt{2}\right]$	$2$
13	${⟨4,3,2⟩}_2$	$\begin{array}{c} 1 2 3 2 2+3+i 3 2+2 \end{array}$	$\mathbb{Z}[i,\sqrt{2}]$	$2$
14	$3\left[8\right]2$	$\begin{array}{c} 1 2 iω22+1-ω 3 3 \end{array}$	$\mathbb{Z}[\omega ,i\sqrt{2}]$	$2$
15	${⟨4,3,2⟩}_6$	$\begin{array}{c} 1 2 3 iω2+1-ω (2+1)iω2+1-ω 2 2iω2+1-ω 3 \end{array}$	$\mathbb{Z}[i,\omega ,\sqrt{2}]$	$2$
16	$5\left[3\right]5$	$\begin{array}{c} 1 2 -η 5 5 \end{array}$	$\mathbb{Z}\left[\eta \right]$	$2$
17	$5\left[6\right]2$	$\begin{array}{c} 1 2 iη3+1-η 5 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/20}\right]$	$2$
18	$5\left[4\right]3$	$\begin{array}{c} 1 2 -ω-η 5 3 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/15}\right]$	$2$
19	${⟨5,3,2⟩}_{30}$	$\begin{array}{c} 1 2 3 iη3+1-η iω2η3+iη3-η -iω2η2-ω iη2ω2(1+η)+1-ω -ω-η 5 3 \end{array}$	$\mathbb{Z}\left[e^{2\pi i/60}\right]$	$2$
20	$3\left[5\right]3$	$\begin{array}{c} 1 2 -ω(η+η4+2) 3 3 \end{array}$	$\mathbb{Z}[\omega ,\frac{1+\sqrt{5}}{2}]$	$2$
21	$3\left[10\right]2$	$\begin{array}{c} 1 2 iω2(η+η4+1)-ω+1 3 \end{array}$	$\mathbb{Z}[\omega ,i\frac{1+\sqrt{5}}{2}]$	$2$
22	${⟨5,3,2⟩}_2$	$\begin{array}{c} 1 2 3 2 (i+1)(η+η4)+4 η+η4+3 3 \end{array}$	$\mathbb{Z}[\frac{\sqrt{5}-1}{2},i\frac{\sqrt{5}-1}{2}]$	$2$
23	$H_3$	$\begin{array}{c} 1 2 3 3+52 \end{array}$	$\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$	$3$
24	$J_3\left(4\right)$	$\begin{array}{c} 1 2 3 2 1+i72 \end{array}$	$\mathbb{Z}\left[\frac{1+i\sqrt{7}}{2}\right]$	$3$
25	$L_3$	$\begin{array}{c} 1 2 3 -ω -ω 3 3 3 \end{array}$	$\mathbb{Z}\left[\omega \right]$	$3$
26	$M_3$	$\begin{array}{c} 1 2 3 -ω 1-ω 3 3 \end{array}$	$\mathbb{Z}\left[\omega \right]$	$3$
27	$J_3\left(5\right)$	$\begin{array}{c} 1 2 3 -2ωcosπ2 4cos2π5 \end{array}$	$\mathbb{Z}[\omega ,\frac{1+\sqrt{5}}{2}]$	$3$
28	$F_4$	$\begin{array}{c} 1 2 3 4 2 \end{array}$	$\mathbb{Z}$	$4$
29	${[2\hspace{0.17em}1;1]}^4=N_4$	$\begin{array}{c} 1 2 3 4 i \end{array}$	$\mathbb{Z}\left[i\right]$	$4$
30	$H_4$	$\begin{array}{c} 1 2 3 4 3+52 \end{array}$	$\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$	$4$
31	$\left[{\left(\frac{1}{2}{\gamma }_3^4\right)}^{+1}\right]=E{N}_4$	$\begin{array}{c} 1 2 2 2 2 3 4 5 i i-1 \end{array}$	$\mathbb{Z}\left[i\right]$	$4$
32	$L_4$	$\begin{array}{c} 1 2 3 4 -ω -ω -ω 3 3 3 3 \end{array}$	$\mathbb{Z}\left[\omega \right]$	$4$
33	${[2\hspace{0.17em}1;2]}^3=K_5$	$\begin{array}{c} 1 2 3 4 5 1+ω \end{array}$	$\mathbb{Z}\left[\omega \right]$	$5$
34	${[2\hspace{0.17em}1;3]}^3=K_6$	$\begin{array}{c} 1 2 3 4 5 6 1+ω \end{array}$	$\mathbb{Z}\left[\omega \right]$	$6$
35	$E_6$	$\begin{array}{c} 1 3 4 5 6 2 \end{array}$	$\mathbb{Z}$	$6$
36	$E_7$	$\begin{array}{c} 1 3 4 5 6 7 2 \end{array}$	$\mathbb{Z}$	$7$
37	$E_8$	$\begin{array}{c} 1 3 4 5 6 7 8 2 \end{array}$	$\mathbb{Z}$	$8$

Remark. For those groups among them that are of the form $\text{Lin}\hspace{0.17em}W,$ where $W$ is an infinite irreducible complex $r\text{-group,}$ one can obtain an explicit expression for the set of lines ${\ell }_{r_j},$ $1\le j\le s,$ from Table 2 below (by taking a vector $e_j$ of unit length in each ${\ell }_{R_r},$ $1\le j\le s\text{).}$ The explicit expressions of this kind for other groups may either be found in [Coh1976] or be derived directly from the graphs.

2) $W$ is infinite

This case was not investigated earlier and is our main concern in these lectures. The results are formulated in the next section. Here we shall give only several trivial examples, which show first of all, that the groups under consideration do exist and, secondly, that we have here a phenomenon which does not occur in the real case.

Examples.

We consider the case where $n=\text{dim}\hspace{0.17em}E=1\text{.}$

Let $a\in E$ be a point. We identify $A\left(E\right)$ and $GL\left(V\right)·V$ by means of ${\kappa }_a,$ see Section 1.1.

Let $\mathrm{\Gamma }\ne 0$ be a lattice in $V\text{.}$ (Here and further on a lattice means a discrete subgroup of the additive group of a vector space, not necessarily of maximal rank.)

Let us consider a subgroup $$W=\left\{(\pm 1,t)\hspace{0.17em}\right|\hspace{0.17em}t\in \mathrm{\Gamma }\}$$ of $GL\left(V\right)·V\text{.}$ This subgroup acts on $E$ (see Section 1.1) and it is easy to check that it is an infinite irreducible complex $r\text{-group.}$

There are two possibilities: either $\text{rk}\hspace{0.17em}\mathrm{\Gamma }=1$ or $\text{rk}\hspace{0.17em}\mathrm{\Gamma }=2\text{.}$ If $\text{rk}\hspace{0.17em}\mathrm{\Gamma }=1$ $$\begin{array}{c} i \end{array}$$ then $E/W$ is not compact, i.e., by definition, $W$ is a noncrystallographic group. In this case, $W$ can be viewed as a real $r\text{-group}$ on $\mathrm{\Gamma }{\otimes }_{\mathbb{Z}}ℝ\text{.}$

If $\text{rk}\hspace{0.17em}\mathrm{\Gamma }=2$ $$\begin{array}{c} 0 \end{array}$$ then $E/W$ is compact, i.e., by definition, $W$ is a crystallographic group.

As we know, it is impossible for an infinite real $r\text{-group}$ to be noncrystallographic, see Section 1.5.

Denote by $\mathrm{\Lambda }$ a lattice of· equilateral triangles in $V$ $\text{(}=ℂ\text{),}$ $$\begin{array}{c}$ 0$\end{array}$$ and by $L$ a lattice of squares in $V$ $\text{(}=ℂ\text{)}$ $$\begin{array}{c} 0 \end{array}$$ Then it is not difficult to check that the groups $$\begin{array}{c}\left\{({\omega }^l,t)\hspace{0.17em}\right|\hspace{0.17em}t\in \mathrm{\Lambda },l\in \mathbb{Z}\},\\ \left\{(\pm {\omega }^l,t)\hspace{0.17em}\right|\hspace{0.17em}t\in \mathrm{\Lambda },l\in \mathbb{Z}\},\end{array}$$ and $$\left\{(i^l,t)\hspace{0.17em}\right|\hspace{0.17em}t\in L,l\in \mathbb{Z}\}$$ are infinite irreducible complex crystallographic $r\text{-groups.}$

Exercise. Prove that all infinite irreducible complex $1\text{-dimensional}$ $r\text{-groups}$ are equivalent to subgroups of those described in these examples.

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