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.

Formulation of the results

We assume in this chapter that $k=ℂ\text{.}$

Let $W$ be an irreducible infinite $r\text{-group,}$ $W\subset A\left(E\right)\text{.}$ As we have seen in the example above, there are two possibilities: either $W$ is noncrystallographic (i.e. $E/W$ is not compact) or $W$ is crystallographic $\text{(}E/W$ is compact). First, we shall describe the structure of noncrystallographic groups. To do this we need an auxiliary construction.

Complexifications and real forms

Let us consider $V$ as a real vector space (of dimension $2n\text{).}$ A linear subspace $V_{ℝ}$ of this real vector space is called a real form of $V$ if

a)	the natural map $$V_{ℝ}{\otimes }_{ℝ}ℂ\to V$$ is an isomorphism, i.e. some (hence, any) $ℝ\text{-basis}$ of $V_{ℝ}$ is a $ℂ\text{-basis}$ of $V\text{;}$
b)	the restriction $⟨\hspace{0.17em}|\hspace{0.17em}⟩{|}_{V_{ℝ}}$ of $⟨\hspace{0.17em}|\hspace{0.17em}⟩$ to $V_{ℝ}$ is realvalued (hence $V_{ℝ}$ is euclidean with respect to $⟨\hspace{0.17em}|\hspace{0.17em}⟩{|}_{V_{ℝ}}\text{).}$

If $V_{ℝ}$ is a real form of $V$ then $V$ is the complexification of $V_{ℝ}\text{.}$

Let $a\in E$ be a point. We can consider $E$ as a real affine space of dimension $2n\text{.}$ The affine subspace $$E_{ℝ}=a+V_{ℝ}$$ of this affine space is called a real form of $E$ and $E$ is called the complexification of $E_{ℝ}\text{.}$

It is clear that every real euclidean linear, resp. affine space is isomorphic to a real form of a certain complex hermitian linear, resp. affine space.

Proposition. One has the following properties:

1)	$U\left(V\right)$ acts transitively on the set of real forms of $V\text{.}$
2)	The group of motions of $E$ acts transitively on the set of real forms of $E\text{.}$
3)	Every motion $\gamma$ of a euclidean affine space $E_{ℝ}$ can be extended in a unique way to a motion ${\gamma }_{ℂ}$ of $E\text{.}$ This motion ${\gamma }_{ℂ}$ is called the complexification of $\gamma$ (and $\gamma$ is called the real form of ${\gamma }_{ℂ}\text{).}$
4)	${\text{dim}}_{ℝ}{H}_{\gamma }={\text{dim}}_{ℂ}{H}_{{\gamma }_{ℂ}}\text{.}$ Specifically, $\gamma$ is a reflection iff ${\gamma }_{ℂ}$ is a reflection.

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

This proposition gives a method for constructing noncrystallographic infinite $r\text{-groups.}$ Indeed, let $G\subset A\left(E_{ℝ}\right)$ be an infinite (real) $r\text{-group.}$ Then it is easy to see that $$G_{ℂ}=\left\{{\gamma }_{ℂ}\hspace{0.17em}\right|\hspace{0.17em}\gamma \in G\}\subset A\left(E\right)$$ is an infinite complex noncrystallographic $r\text{-group}$ (and $G_{ℂ}$ is irreducible if and only if $G$ is).

Classification of infinite irreducible complex noncrystallographic $r\text{-groups:}$ the result

It appears that the construction above leads to any such group. More precisely, one has the following theorem (see also Section 1.5,2)):

Theorem. Let $W$ be an infinite irreducible complex $r\text{-group.}$ Then $W$ is noncrystallographic if and only if it is equivalent to the complexification of an irreducible affine Weyl group.

		Proof.
	Proof is given in Section 3.4. $\square$

The description of crystallographic groups is much more complicated. In order to give this description we need some preparations and extra notation.

Ingredients of the description

The subgroup of translations in $W$ will be denoted by $\text{Tran}\hspace{0.17em}W\text{.}$ $$\text{Tran}\hspace{0.17em}W=W\cap \text{Tran}\hspace{0.17em}A\left(E\right),$$ cf. Section 1.1. It is clear that $\hspace{0.17em}W⊲W$ and $$W/\text{Tran}\hspace{0.17em}W\cong \text{Lin}\hspace{0.17em}W\text{.}$$ We usually identify $\text{Tran}\hspace{0.17em}W$ with a subgroup of the additive group of $V$ by means of the map ${\gamma }_v\mapsto v\text{.}$ Clearly this subgroup is a $\text{Lin}\hspace{0.17em}W\text{-invariant}$ lattice in $V\text{.}$

It will be proven in Section 3.1 that $\text{Tran}\hspace{0.17em}W$ is a lattice of full rank (i.e. of rank $2n\text{)}$ and $\text{Lin}\hspace{0.17em}W$ is a finite group (hence, $\text{Lin}\hspace{0.17em}W$ is a finite irreducible complex linear $r\text{-group,}$ see 1.4). Therefore, to describe $W,$ one needs to point out a group $\text{Lin}\hspace{0.17em}W\subset GL\left(V\right)$ from the Shephard and Todd list (i.e. from the theorem in 1.6), a $\text{Lin}\hspace{0.17em}W\text{-invariant}$ lattice $\text{Tran}\hspace{0.17em}W\subset V$ of rank $2n$ and the way $\text{Lin}\hspace{0.17em}W$ and $\text{Tran}\hspace{0.17em}W$ are "glued" together. This is done below as follows:

1) Lin $W$ is given by its graph as in Section 1.6.

2) $\text{Tran}\hspace{0.17em}W$ is described explicitly by linear combinations of vectors $e_j,$ $1\le j\le s,$ that generate $\text{Tran}\hspace{0.17em}W\text{.}$ Here $R_j=R_{e_j,{\theta }_j},$ $1\le j\le s,$ is a fixed generating system of reflections of $\text{Lin}\hspace{0.17em}W$ which is related to the graph of $\text{Lin}\hspace{0.17em}W$ given in 1) as described in 1.6. To point out the vectors $e_j,$ $1\le j\le s,$ explicitly, we assume that $V$ is a subspace of a standard hermitian infinitedimensional coordinate space ${ℂ}^{\infty }$ i.e. the space, whose elements are the sequences $(a_1,a_2,\dots )$ with only a finite number of nonzero elements $a_j,$ and a scalar product defined by the formula $$⟨(a_1,a_2,\dots )\hspace{0.17em}|\hspace{0.17em}(b_1,b_2,\dots )⟩=\sum _{j=1}^{\infty }{a}_j{\bar{b}}_j\text{.}$$ The vectors $e_j,$ $1\le j\le s,$ are given by their, coordinates on a standard basis ${\epsilon }_1,{\epsilon }_2,\dots$ of ${ℂ}^{\infty },$ where $${\epsilon }_j=(0,\dots ,0,1,0,\dots )\text{.}$$

3) The problem how to describe the "glueing" of $\text{Lin}\hspace{0.17em}W$ and $\text{Tran}\hspace{0.17em}W$ comes down to the determination of an extension of $\text{Tran}\hspace{0.17em}W$ by $\text{Lin}\hspace{0.17em}W,$ $$0\to \text{Tran}\hspace{0.17em}W\to W\to \text{Lin}\hspace{0.17em}W\to 1\text{.}$$ Therefore it is done by means of cohomology. Let us show how it can be done.

Cohomology

Let $G$ be a subgroup of $A\left(E\right)$ and write $T=\text{Tran}\hspace{0.17em}G,$ $K=\text{Lin}\hspace{0.17em}G\text{.}$ Choose a point $a\in E\text{.}$ Take $P\in K$ and let $\gamma \in G$ be such that $\text{Lin}\hspace{0.17em}\gamma =P\text{.}$ We have $${\kappa }_a\left(\gamma \right)=(P,s\left(P\right)),s\left(P\right)\in V\text{.}$$ It is easy to see that the map $$\bar{s}:K\to V/T,\bar{s}\left(P\right)=s\left(P\right)+T,$$ is well defined and is in fact a $1\text{-cocycle,}$ i.e. $$\bar{s}\left(PQ\right)=\bar{s}\left(P\right)+P\bar{s}\left(Q\right),P,Q\in K\text{.}$$ (here $K$ acts on $V/T$ in the natural way).

Vice versa, if $$\bar{r}:K\to V/T$$ is an arbitrary $1\text{-cocycle,}$ let us consider an arbitrary map $$r:K\to V$$ such that $$\bar{r}\left(P\right)=r\left(P\right)+T,P\in K\text{.}$$ Then the set $$\left\{(P,r\left(P\right)+t)\hspace{0.17em}\right|\hspace{0.17em}t\in T,P\in K\}$$ is a subgroup $H$ of $A\left(E\right)$ with $\text{Lin}\hspace{0.17em}H=K$ and $\text{Tran}\hspace{0.17em}H=T\text{.}$

If we replace $a$ by an other point $b\in E,$ then (see 1.1). $${\kappa }_b\left(\gamma \right)={\kappa }_a\left({\gamma }_{a-b}\gamma {\gamma }_{b-a}\right)=(P,s\left(P\right)+\underset{\underset{1\text{-coboundary}}{⏟}}{v-Pv}),\text{where}\hspace{0.17em}v=a-b\text{.}$$ Therefore we have a bijection between the set of $\text{Tran}\hspace{0.17em}A\left(E\right)\text{-conjugacy}$ classes of subgroups $G$ of $A\left(E\right)$ with $\text{Lin}\hspace{0.17em}G=K,$ $\text{Tran}\hspace{0.17em}G=T$ and the group $H^1(K,V/T)\text{.}$

However we have to consider subgroups of $A\left(E\right)$ up to equivalence, i.e. up to $A\left(E\right)\text{-conjugation}$ (and not just up to $\text{Tran}\hspace{0.17em}A\left(E\right)\text{-conjugation)!}$ This can be done as follows by means of an extra relation on $H^1(K,V/T)\text{.}$ Let $$N(K,T)=\{Q\in GL\left(V\right)\hspace{0.17em}|\hspace{0.17em}QK{Q}^{-1}=K,QT=T\}\text{.}$$ If $Q\in N(K,T)$ and $\bar{s}:K\to V/T$ is a $1\text{-cocycle,}$ resp. $1\text{-coboundary,}$ then it is easy to check that the map $$Q\left(\bar{s}\right):K\to V/T$$ given by the formula $$Q\left(\bar{s}\right)\left(P\right)=Q\bar{s}\left(Q^{-1}PQ\right),P\in K,$$ is again a $1\text{-cocycle,}$ resp. $1\text{-coboundary}$ (here $Q$ acts on $V/T$ in the natural way). Therefore we have an action of $N(K,T)$ on $H^1(K,V/T)$ (clearly, by means of automorphisms).

Let $\delta \in A\left(E\right)$ be such that $\text{Lin}\hspace{0.17em}\delta G{\delta }^{-1}=K,$ $\text{Tran}\hspace{0.17em}\delta G{\delta }^{-1}=T\text{.}$ We want to calculate the cocycle that corresponds to $\delta G{\delta }^{-1}\text{.}$ Changing $\delta$ to $\delta {\gamma }_v,$ where $v=\left(\text{Lin}\hspace{0.17em}{\delta }^{-1}\right)(a-\delta \left(a\right)),$ we can assume that ${\kappa }_a\left(\delta \right)=(Q,0),$ $Q\in N(K,T)\text{.}$ Let $P\in K$ and $\lambda \in G$ be such that ${\kappa }_a\left(\lambda \right)=(Q^{-1}PQ,s\left(Q^{-1}PQ\right))\text{.}$ Then ${\kappa }_a\left(\delta \lambda {\delta }^{-1}\right)=(P,Qs\left(Q^{-1}PQ\right))\text{.}$ Therefore the cocycle corresponding to $\delta G{\delta }^{-1}$ is $Q\left(\bar{s}\right)$ where $\bar{s}$ is the cocycle corresponding to $G\text{.}$

We see now that there is a bijection between the set of classes of equivalent subgroups $G\subset A\left(E\right)$ with $\text{Lin}\hspace{0.17em}G=K,$ $\text{Tran}\hspace{0.17em}G=T$ and the set of $N(K,T)\text{-orbits}$ in $H^1(K,V/T)\text{.}$

With all these facts in mind, we determine the extension $W$ (of $\text{Tran}\hspace{0.17em}W$ by $\text{Lin}\hspace{0.17em}W\text{)}$ by pointing out a $1\text{-cocycle}$ which represents the corresponding element of $H^1(\text{Lin}\hspace{0.17em}W,V/\text{Tran}\hspace{0.17em}W)$ (in fact, the whole $N(\text{Lin}\hspace{0.17em}W,\text{Tran}\hspace{0.17em}W)\text{-orbit}$ in $H^1(\text{Lin}\hspace{0.17em}W,V/\text{Tran}\hspace{0.17em}W)\text{).}$ In order to do so, we need only give the values of this $1\text{-cocycle}$ on the elements of a generating system of reflections of $\text{Lin}\hspace{0.17em}W\text{.}$ Technically it is more convenient to realize it as follows.

Let $\widetilde{\text{Lin}\hspace{0.17em}W}$ be a free group with generators $r_j,$ $1\le j\le s\text{.}$ We have an epimorphism $\varphi :\widetilde{\text{Lin}\hspace{0.17em}W}\to \text{Lin}\hspace{0.17em}W,$ $\varphi \left(r_j\right)=R_j,$ $1\le j\le s\text{.}$ The kernel of $\varphi$ is the subgroup of "relations" of $\text{Lin}\hspace{0.17em}W\text{.}$ This epimorphism leads in a natural way to an action of $\widetilde{\text{Lin}\hspace{0.17em}W}$ on $V\text{.}$ A $1\text{-cocycle}$ $c$ of $\widetilde{\text{Lin}\hspace{0.17em}W}$ with values in $V$ is given by its values on the generators $r_j,$ $$c\left(r_j\right),1\le j\le s,$$ and these values may be arbitrary (because $\widetilde{\text{Lin}\hspace{0.17em}W}$ is free). It is easy to see that the formula $$R_j\to c\left(r_j\right)+\text{Tran}\hspace{0.17em}W,1\le j\le s,$$ defines a $1\text{-cocycle}$ of $\text{Lin}\hspace{0.17em}W$ with values in $V/\text{Tran}\hspace{0.17em}W$ iff $c\left(F\right)\in \text{Tran}\hspace{0.17em}W$ for every $F\in \text{Ker}\hspace{0.17em}\varphi \text{.}$ It is also clear that every $1\text{-cocycle}$ of $\text{Lin}\hspace{0.17em}W$ with values in $V/\text{Tran}\hspace{0.17em}W$ is obtained in such a way.

We shall give the extension $W$ (of $\text{Tran}\hspace{0.17em}W$ by $\text{Lin}\hspace{0.17em}W\text{)}$ by writing down the vectors $c\left(r_j\right),$ $1\le j\le s\text{.}$

We are now ready to formulate the results of the classification of infinite irreducible crystallographic $r\text{-groups.}$

Denote by $K_b$ the finite linear irreducible $r\text{-group}$ which has the number $b$ in the list of Shephard and Todd (i.e. in the first column of Table 1. This in spite of the slight confusion with Cohen's notation $K_5,K_6\text{).}$

Description of the group of linear parts: the result.

First of all, there is an analogue of the theorem of Section 1.5.

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

a)	There exists a nonzero $K\text{-invariant}$ lattice in $V\text{.}$
b)	There exists a $K\text{-invariant}$ lattice of rank $2n$ in $V\text{.}$
c)	$K=\text{Lin}\hspace{0.17em}W$ where $W$ is an infinite crystallographic $r\text{-group.}$
d)	The ring with unity, generated over $\mathbb{Z}$ by all cyclic products of a graph of $K,$ lies in the ring of algebraic integers of a purely imaginary quadratic extension of $\mathbb{Q}\text{.}$
e)	$K$ is defined over a purely imaginary quadratic extension of $\mathbb{Q}\text{.}$
f)	$K$ is one of the groups: $$\begin{array}{c}{K}_1\text{;}{K}_2\hspace{0.17em}(m=2,3,4,6)\text{;}{K}_3\hspace{0.17em}(m=2,3,4,6)\text{;}\\ K_4\text{;}{K}_5\text{;}{K}_8\text{;}{K}_{12}\text{;}{K}_{24}\text{;}{K}_{25}\text{;}{K}_{26}\text{;}{K}_{28}\text{;}{K}_{29}\text{;}\\ K_{31}\text{;}{K}_{32}\text{;}{K}_{33}\text{;}{K}_{34}\text{;}{K}_{35}\text{;}{K}_{36}\text{;}{K}_{37}\text{.}\end{array}$$

		Proof.
	Proof is given in Section 4.6. $\square$

Now we shall describe the crystallographic groups themselves.

The list of irreducible infinite crystallographic complex groups.

This list is given in the following theorem (we use the notation: $\mathrm{\Omega }=\{z\in ℂ\hspace{0.17em}|\hspace{0.17em}-\frac{1}{2}\le \text{Re}\hspace{0.17em}z<\frac{1}{2},\left|z\right|\ge 1$ if $\text{Re}\hspace{0.17em}z\le 0$ and $\left|z\right|>1$ if $\text{Re}\hspace{0.17em}z>0\}$ - this is the "modular strip"; $[\alpha ,\beta ]=\{a\alpha +b\beta \hspace{0.17em}|\hspace{0.17em}a,b\in \mathbb{Z}\}$ for arbitrary $\alpha ,\beta \in ℂ\text{).}$

Theorem. The following list is the complete list of irreducible infinite crystallographic complex $r\text{-groups}$ $W$ (considered up to equivalence).

The proof is given in the subsequent chapters.

Table 2
The irreducible infinite crystallographic complex $r\text{-groups.}$
Notation of $W$	$n=\text{dim}\hspace{0.17em}W$	$\text{Lin}\hspace{0.17em}W$	$\text{Tran}\hspace{0.17em}W$	$e_1,\dots ,e_s$	cocycle $c$
${\left[A_s\right]}^{\alpha }$ $s\ge 1$	$s$	$K_1,$ type $A_s$ $s\ge 1$	$[1,\alpha ]e_1+\dots +[1,\alpha ]e_s,$ $\alpha \in \mathrm{\Omega }$	$e_j=({\epsilon }_j-{\epsilon }_{j+1})/\sqrt{2}$ $j=1,\dots ,s$	$c=0$
${\left[G(2,1,s)\right]}_1^{\alpha }$ $s\ge 3$	$s$	$K_2$ type $G(2,1,s)$ $s\ge 3$	$[1,\alpha ]e_1+[1,\alpha ]\sqrt{2}{e}_2+\dots +[1,\alpha ]\sqrt{2}{e}_s,$ $\alpha \in \mathrm{\Omega }$	$e_1={\epsilon }_1,$ $e_j=({\epsilon }_{j-1}-{\epsilon }_j)/\sqrt{2}$ $j=2,\dots ,s$
${\left[G(2,1,s)\right]}_2^{\beta }$ $s\ge 3$			$[1,\beta ]e_1+[1,\frac{1+\beta }{2}]\sqrt{2}{e}_2+\cdots +[1,\frac{1+\beta }{2}]\sqrt{2}{e}_s,$ $\beta \in \mathrm{\Omega }$
${\left[G(2,1,s)\right]}_3^{\gamma }$ $s\ge 3$			$[1,\gamma ]e_1+[\frac{1}{2},\gamma ]\sqrt{2}{e}_2+\dots +[\frac{1}{2},\gamma ]\sqrt{2}{e}_s,$ $\gamma \in \mathrm{\Omega }$
${\left[G(2,1,s)\right]}_4^{\delta }$ $s\ge 3$			$\delta \in \mathrm{\Omega }$
${\left[G(2,1,s)\right]}_5^{\lambda }$ $s\ge 3$			$[1,\lambda ]e_1+[\frac{1}{2},\frac{\lambda }{2}]\sqrt{2}{e}_2+\dots +[\frac{1}{2},\frac{\lambda }{2}]\sqrt{2}{e}_s,$ $\lambda \in \mathrm{\Omega }$
${\left[G(3,1,s)\right]}_1$ $s\ge 2$	$s$	$K_2$ type $G(3,1,s)$ $s\ge 2$	$[1,\omega ]e_1+[1,\omega ]\sqrt{2}{e}_2+\dots +[1,\omega ]\sqrt{2}{e}_s$
${\left[G(3,1,s)\right]}_2$ $s\ge 2$			$[1,\omega ]e_1+[1,\omega ]i\sqrt{\frac{2}{3}}{e}_2+\dots +[1,\omega ]i\sqrt{\frac{2}{3}}{e}_s$
${\left[G(4,1,s)\right]}_1$ $s\ge 2$	$s$	$K_2$ type $G(4,1,s)$ $s\ge 2$	$[1,i]e_1+[1,i]\sqrt{2}{e}_2+\dots +[1,i]\sqrt{2}{e}_s$
${\left[G(4,1,s)\right]}_2$ $s\ge 2$			$[1,i]e_1+[1,i]\epsilon e_2+\dots +[1,i]\epsilon e_s$
$\left[G(6,1,s)\right]$ $s\ge 2$	$s$	$K_2$ type $G(6,1,s)$ $s\ge 2$	$[1,\omega ]e_1+[1,\omega ]\sqrt{2}{e}_2+\dots +[1,\omega ]\sqrt{2}{e}_s$
${\left[G(2,2,s)\right]}^{\alpha }$ $s\ge 3$	$s$	$K_2$ $G(2,2,s)$ $s\ge 3$	$[1,\alpha ]e_1+\dots +[1,\alpha ]e_s,$ $\alpha \in \mathrm{\Omega }$	$e_1=-({\epsilon }_1+{\epsilon }_2)/\sqrt{2},$ $e_j=({\epsilon }_{j-1}-{\epsilon }_j)/\sqrt{2},$ $j=2,\dots ,s$
$\left[G(3,3,s)\right]$ $s\ge 3$	s	$K_2,$ type $G(3,3,s)$ $s\ge 3$	$[1,\omega ]e_1+\dots +[1,\omega ]e_s$	$e_1=\omega {\epsilon }_1-{\epsilon }_2,$ $e_j=({\epsilon }_{j-1}-{\epsilon }_j)/\sqrt{2},$ $j=2,\dots ,s$
$\left[G(4,4,s)\right]$ $s\ge 3$	$s$	$K_2$ type $G(4,4,s)$ $s\ge 3$	$[1,i]e_1+\dots +[1,i]e_s$	$e_1=(i{\epsilon }_1-{\epsilon }_2)/\sqrt{2},$ $e_j=({\epsilon }_{j-1}-{\epsilon }_j)/\sqrt{2},$ $j=2,\dots ,s$
$\left[G(6,6,s)\right]$ $s\ge 3$	$s$	$K_2,$ type $G(4,4,s)$ $s\ge 3$	$[1,\omega ]e_1+\dots +[1,\omega ]e_s$	$e_1=((1+\omega )e_1-e_2)/\sqrt{2}$ $e_j=\left({\epsilon }_{j-1}{\epsilon }_j\right)/\sqrt{2}$ $j=2,\dots ,s$
${\left[G(2,1,2)\right]}_1^{\alpha }$	$2$	$K_2,$ type $G(2,1,2)$ $=$ type $G(4,4,2)$	$[1,\alpha ]e_1+[1,\alpha ]\sqrt{2}{e}_2,$ $\alpha \in \mathrm{\Omega }$	$e_1={\epsilon }_1$ $e_2=({\epsilon }_1-{\epsilon }_2)/\sqrt{2}$
${\left[G(2,1,2)\right]}_2^{\beta }$			$[1,\beta ]e_1+[1,\frac{\beta }{2}]\sqrt{2}{e}_2,$ $\beta \in \mathrm{\Omega }$
${\left[G(2,1,2)\right]}_3^{\gamma }$			$[1,\gamma ]e_1+[1,\frac{1+\gamma }{2}]\sqrt{2}{e}_2,$ $\gamma \in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_1^{\alpha }$	2	$K_2,$ type $G(6,6,2)$	$[1,\alpha ]e_1+[1,\alpha ](2+\omega )e_2,$ $\alpha \in \mathrm{\Omega }$	$e_1=((1+\omega ){\epsilon }_1-{\epsilon }_2)/\sqrt{2}$ $e_2=({\epsilon }_1-{\epsilon }_2)/\sqrt{2}$
${\left[G(6,6,2)\right]}_2^{\beta }$			$[1,\beta ]e_1+[1,\frac{\beta }{3}](2+\omega )e_2,$ $\beta \in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_3^{\gamma }$			$[1,\gamma ]e_1+[1,\frac{1+\gamma }{3}](2+\omega )e_2,$ $\gamma \in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_4^{\delta }$			$[1,\delta ]e_1+[1,\frac{2+\delta }{3}](2+\omega )e_2,$ $\delta \in \mathrm{\Omega }$
${\left[G(4,2,s-1)\right]}_1$ $s\ge 3$	$s-1$	$K_2,$ type $G(4,2,s-1)$ $s\ge 3$	$T=[1,i]e_1+\dots +[1,i]e_{s-1}$	$e_1=(i{\epsilon }_1-{\epsilon }_2)/\sqrt{2}$ $e_j=\left({\epsilon }_{j-1}{\epsilon }_j\right)/\sqrt{2}$ $j=2,\dots ,s-1$ $e_s={\epsilon }_{s-1}$
${\left[G(4,2,s-1)\right]}_1^{*}$ $s\ge 3$					$c\left(r_j\right)=0,$ $j=1,\dots ,s-1$ $c\left(r_s\right)=e_s/\sqrt{2}$
${\left[G(4,2,s-1)\right]}_2$ $s\ge 3$			$T\cup (T+\frac{1+i}{2}(e_1+e_2))=$ $[1,i]e_1+\dots +[1,i]e_{s-1}+\frac{1}{\sqrt{2}}[1,i]e_s$		$c=0$
${\left[G(4,2,2)\right]}_3$	$2$	$K_2,$ type $G(4,2,2)$	$[1,i]e_1+[1,i](1+i)e_2$
${\left[G(6,2,s-1)\right]}_1$ $s\ge 3$	$s-1$	$K_2$ type $G(6,2,s-1)$ $s\ge 3$	$[1,\omega ]e_1+\dots +[1,\omega ]e_{s-1}$	$e_1=((1+\omega ){\epsilon }_1-{\epsilon }_2)/\sqrt{2},$ $e_j=({\epsilon }_{j-1}-{\epsilon }_j)/\sqrt{2},$ $j=2,\dots ,s-1$ $e_s={\epsilon }_{s-1}$
${\left[G(6,2,2)\right]}_2$	$2$	$K_2,$ type $G(6,2,2)$	$[1,\omega ]e_1+[1,\omega ](2+\omega )e_2$
${\left[G(6,3,s-1)\right]}_1$ $s\ge 3$	$s-1$	$K_2,$ type $G(6,3,s-1)$ $s\ge 3$	$[1,\omega ]e_1+\dots +[1,\omega ]e_{s-1}$
${\left[G(6,3,2)\right]}_2$	$2$	$K_2,$ type $G(6,3,2)$	$[1,2\omega ]e_1+[2,\omega ]e_2$
$\left[K_3\left(3\right)\right]$	$1$	$K_3$ $m=3$	$[1,\omega ]e_1$	$e_1={\epsilon }_1$
$\left[K_3\left(4\right)\right]$		$K_3$ $m=4$	$[1,i]e_1$
$\left[K_3\left(6\right)\right]$		$K_3,$ $m=6$	$[1,\omega ]e_1$
$\left[K_4\right]$	$2$	$K_4$	$[1,\omega ]e_1+[1,\omega ]e_2$	$e_1={\epsilon }_1,$ $e_2=\frac{1-\omega }{3}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3)$
$\left[K_5\right]$		$K_5$	$[1,\omega ]e_1+[1,\omega ]\sqrt{2}{e}_2$	$e_1={\epsilon }_1,$ $e_2=\frac{1-\omega }{3}(\sqrt{2}{\epsilon }_1+{\epsilon }_2)$
[K8]		$K_8$	$[1,i]e_1+[1,i]e_2$	$e_1={\epsilon }_1,$ $e_2=\frac{1-i}{2}({\epsilon }_1-{\epsilon }_2)$
[K12]		$K_{12}$	$[1,i\sqrt{2}]e_1+[1,i\sqrt{2}]e_2$	$e_1=\frac{1}{\sqrt{2}}{\epsilon }_1+\frac{1+i}{2}{\epsilon }_2,$ $e_2=\frac{\sqrt{2}+(\sqrt{2}-2)i}{4}{\epsilon }_1+\frac{2+\sqrt{2}-\sqrt{2}i}{4}{\epsilon }_2,$ $e_3=\frac{1}{\sqrt{2}}{\epsilon }_1+\frac{1-i}{2}{\epsilon }_2$
[K12]*					$c\left(r_1\right)=c\left(r_2\right)=0$ $c\left(r_3\right)=\frac{1+i}{2}{e}_3$
[K24]	3	$K_{24}$	$[1,\frac{1+i\sqrt{7}}{2}]e_1+[1,\frac{1+i\sqrt{7}}{2}]e_2+[1,\frac{1+i\sqrt{7}}{2}]e_3$	$e_1={\epsilon }_2,$ $e_2=(1-i\sqrt{7})({\epsilon }_2+{\epsilon }_3)/4,$ $e_3=(-{\epsilon }_1-{\epsilon }_2+\frac{1+i\sqrt{7}}{2}{\epsilon }_3)/2$	$c=0$
$\left[K_{25}\right]$		$K_{25}$	$[1,\omega ]e_1+[1,\omega ]e_2+[1,\omega ]e_3$	$e_1={\epsilon }_3,$ $e_2=\frac{1-\omega }{3}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3),$ $e_3=-\omega {\epsilon }_2$
${\left[K_{26}\right]}_1$		$K_{26}$	$[1,\omega ]e_1+[1,\omega ]e_2+[1,\omega ]\sqrt{2}{e}_3$	$e_1=\frac{1-{\omega }^2}{3}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3),$ $e_2={\epsilon }_3,$ $e_3=\frac{1}{\sqrt{2}}({\epsilon }_2-{\epsilon }_3)$
${\left[K_{26}\right]}_2$			$[1,\omega ]e_1+[1,\omega ]e_2+[1,\omega ]o\sqrt{\frac{2}{3}}{e}_3$
${\left[F_4\right]}_1^{\alpha }$	$4$	$K_{28}$ $F_4$	$[1,\alpha ]e_1+[1,\alpha ]e_2+[1,\alpha ]\sqrt{2}{e}_3+[1,\alpha ]\sqrt{2}{e}_4,$ $\alpha \in \mathrm{\Omega }$	$e_j=({\epsilon }_{j+1}-{\epsilon }_{j+2})/\sqrt{2},$ $j=1,2,$ $e_3={\epsilon }_4,$ $e_4=({\epsilon }_1-{\epsilon }_2-{\epsilon }_3-{\epsilon }_4)/2$
${\left[F_4\right]}_2^{\beta }$			$[1,\beta ]e_1+[1,\frac{\beta }{2}]e_2+[1,\beta ]\sqrt{2}{e}_3+[1,\frac{\beta }{2}]\sqrt{2}{e}_4,$ $\beta \in \mathrm{\Omega }$
${\left[F_4\right]}_3^{\gamma }$			$[1,\gamma ]e_1+[1,\gamma ]e_2+[1,\frac{1+\gamma }{2}]\sqrt{2}{e}_3+$ $[1,\frac{1+\gamma }{2}]\sqrt{2}{e}_4,$ $\gamma \in \mathrm{\Omega }$
$\left[K_{29}\right]$		$K_{29}$	$[1,i]e_1+[1,i]e_2+[1,i]e_3+[1,i]e_4$	$e_1=\frac{1}{\sqrt{2}}({\epsilon }_2-{\epsilon }_4)$ $e_2=\frac{1}{\sqrt{2}}(-i{\epsilon }_2+{\epsilon }_3)$ $e_3=\frac{1}{\sqrt{2}}(-{\epsilon }_3+{\epsilon }_4)$ $e_4=\frac{-1+i}{2\sqrt{2}}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3+{\epsilon }_4)$
$\left[K_{31}\right]$		$K_{31}$	$[1,i]e_1+[1,i]e_2+[1,i]e_3+[1,i]e_4$	$e_1=\frac{1}{\sqrt{2}}({\epsilon }_2-{\epsilon }_4)$ $e_2=\frac{1}{\sqrt{2}}(-i{\epsilon }_2+{\epsilon }_3)$ $e_3=\frac{1}{\sqrt{2}}(-{\epsilon }_3+{\epsilon }_4)$ $e_4=\frac{-1+i}{2\sqrt{2}}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3+{\epsilon }_4)$ $e_5=\frac{1-i}{\sqrt{2}}{\epsilon }_4$
${\left[K_{31}\right]}^{*}$					$c\left(r_j\right)=0$ $j=1,2,3,4,$ $c\left(r_5\right)=\frac{1+i}{2}{e}_5$
$\left[K_{32}\right]$		$K_{32}$	$[1,\omega ]e_1+[1,\omega ]e_2+[1,\omega ]e_3+[1,\omega ]e_4$	$e_1={\epsilon }_3,$ $e_2=\frac{1-\omega }{3}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3),$ $e_3=-\omega {\epsilon }_2,$ $e_4=\frac{{\omega }^2-\omega }{3}(-{\epsilon }_1+{\epsilon }_2+{\epsilon }_4)$	$c=0$
$\left[K_{33}\right]$	$5$	$K_{33}$	$[1,\omega ]e_1+\dots +[1,\omega ]e_n$	$e_1=\frac{\omega }{\sqrt{2}}({\epsilon }_5+{\epsilon }_6),$ $e_2=-\frac{\omega }{2\sqrt{2}}(-{\epsilon }_1+(1+2\omega ){\epsilon }_2$ $+{\epsilon }_3+{\epsilon }_4+{\epsilon }_5+{\epsilon }_6),$ $e_j=\frac{1}{\sqrt{2}}({\epsilon }_{j-2}-{\epsilon }_{j-1}),$ $j=3,4,\dots ,n$
$\left[K_{23}\right]$	$6$	$K_{24}$
${\left[E_6\right]}^{\alpha }$		$K_{35},$ $E_6$	$[1,\alpha ]e_1+\dots +[1,\alpha ]e_n,$ $\alpha \in \mathrm{\Omega }$	$e_1=({\epsilon }_1-{\epsilon }_2-{\epsilon }_3-{\epsilon }_4$ $-{\epsilon }_5-{\epsilon }_6-{\epsilon }_7+{\epsilon }_8)/2\sqrt{2},$ $e_2=({\epsilon }_1+{\epsilon }_2)/\sqrt{2},$ $e_j=(-{\epsilon }_{j-2}+{\epsilon }_{j-1})/\sqrt{2},$ $j=3,\dots ,n\text{.}$
${\left[E_7\right]}^{\alpha }$	$7$	$K_{36},$ $E_7$
${\left[E_8\right]}^{\alpha }$	$8$	$K_{37},$ $E_8$

Equivalence

Theorem. The following list is the complete list of groups $W$ and $W\prime ,$ $W\ne W\prime ,$ from Table 2 which are equivalent:

Table 3
Pairs of equivalent irreducible infinite crystallographic complex $r\text{-groups}$
$W$	$W\prime$	condition
${\left[G(2,1,s)\right]}_2^{1+\omega },$ $s\ge 3$	${\left[G(2,1,s)\right]}_3^{1+\omega },$ $s\ge 3$	--
${\left[G(2,1,s)\right]}_2^{1+\omega },$ $s\ge 3$	${\left[G(2,1,s)\right]}_4^{1+\omega },$ $s\ge 3$	--
${\left[G(2,1,s)\right]}_3^i,$ $s\ge 3$	${\left[G(2,1,s)\right]}_4^i,$ $s\ge 3$	--
${\left[G(2,1,2)\right]}_2^{\beta }$	${\left[G(2,1,2)\right]}_2^{-2/\beta }$	$-2/\beta \in \mathrm{\Omega }$
${\left[G(2,1,2)\right]}_2^{1+\omega }$	${\left[G(2,1,2)\right]}_3^{1+\omega }$	--
${\left[G(2,1,2)\right]}_2^{\beta }$	${\left[G(2,1,2)\right]}_3^{1-2/\beta }$	$1-2/\beta \in \mathrm{\Omega }$
${\left[G(2,1,2)\right]}_2^{\gamma }$	${\left[G(2,1,2)\right]}_3^{(\gamma -1)/(\gamma +1)}$	$(\gamma -1)/(\gamma +1)\in \mathrm{\Omega }$
${\left[G(2,1,2)\right]}_2^{\beta }$	${\left[G(2,1,2)\right]}_3^{-1-2/\beta }$	$-1-2/\beta \in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_2^{\beta }$	${\left[G(6,6,2)\right]}_2^{-3/\beta }$	$-3/\beta \in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_3^{\gamma }$	${\left[G(6,6,2)\right]}^{(2\gamma -1)/(\gamma +1)}$	$(2\gamma -1)/(\gamma +1)\in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_2^{\beta }$	${\left[G(6,6,2)\right]}^{-1+3/\beta }$	$-1+3/\beta \in \mathrm{\Omega }$
${\left[G(6,6,2)\right]}_2^{\beta }$	${\left[G(6,6,2)\right]}_3^{2-3/\beta }$	$2-3/\beta \in \mathrm{\Omega }$
${\left[F_4\right]}_2^{\beta }$	${\left[F_4\right]}_2^{-2/\beta }$	$-2/\beta \in \mathrm{\Omega }$
${\left[F_4\right]}_2^{1+\omega }$	${\left[F_4\right]}_3^{1+\omega }$	--
${\left[F_4\right]}_2^{\beta }$	${\left[F_4\right]}_3^{1-2/\beta }$	$1-2/\beta \in \mathrm{\Omega }$
${\left[F_4\right]}_3^{\gamma }$	${\left[F_4\right]}_3^{(\gamma -1)/(\gamma +1)}$	$(\gamma -1)/(\gamma +1)\in \mathrm{\Omega }$
${\left[F_4\right]}_2^{\beta }$	${\left[F_4\right]}_3^{-1-2/\beta }$	$-1-2/\beta \in \mathrm{\Omega }$

Proof of this theorem is rather technical and will not be given here.

The structure of an extension of $\text{Tran}\hspace{0.17em}W$ by $\text{Lin}\hspace{0.17em}W\text{.}$

As we have seen in Section 1.5, if $k=ℝ$ then the structure of an infinite irreducible $r\text{-group}$ $W$ as an extension of $\text{Tran}\hspace{0.17em}W$ by $\text{Lin}\hspace{0.17em}W$ is very simple: it is always a semidirect product. The situation is more complicated when $k=ℂ,$ because there exist infinite irreducible complex crystallographic $r\text{-groups}$ $W$ which are not semidirect products of $\text{Tran}\hspace{0.17em}W$ and $\text{Lin}\hspace{0.17em}W\text{.}$

Theorem. The groups $W$ from Table 2 which are not semidirect products of $\text{Tran}\hspace{0.17em}W$ and $\text{Lin}\hspace{0.17em}W$ are $${\left[G(4,2,s)\right]}_1^{*},\hspace{0.17em}{\left[K_{12}\right]}^{*}\hspace{0.17em}\text{and}\hspace{0.17em}{\left[K_{31}\right]}^{*}\text{.}$$

Theorem. Let $K\subset GL\left(V\right)$ be a finite irreducible $r\text{-group}$ and let $T\subset V$ be a $K\text{-invariant}$ lattice. Assume that there exists a crystallographic $r\text{-group}$ $W$ with $\text{Lin}\hspace{0.17em}W=K,$ $\text{Tran}\hspace{0.17em}W=T\text{.}$ Then the set of those elements of $H^1(K,V/T)$ which correspond to such subgroups $W$ is in fact a subgroup of $H^1(K,V/T)$ and the order of this subgroup is $\le 2\text{.}$

The rings and fields of definition of $\text{Lin}\hspace{0.17em}W\text{.}$

As we have seen in Section 1.5, if $k=ℝ$ then the group $\text{Lin}\hspace{0.17em}W$ for an infinite irreducible $r\text{-group}$ $W$ is defined over $\mathbb{Q}\text{.}$ If $k=ℂ$ then $\text{Lin}\hspace{0.17em}W$ for an infinite irreducible crystallographic $r\text{-group}$ $W$ is defined over a certain purely imaginary quadratic extension of $\mathbb{Q},$ see the theorem in Section 2.5. We can describe this extension precisely.

Theorem. Let $K\subset GL\left(V\right)$ be a finite irreducible complex $r\text{-group.}$ Then the ring with unity generated over $\mathbb{Z}$ by the set of all cyclic products related to an arbitrary fixed generating system of reflections of $K$ coincides with the ring $\mathbb{Z}\left[\text{Tr}K\right]$ generated over $\mathbb{Z}$ by the set of traces of all elements of $K\text{.}$ The ring $\mathbb{Z}\left[\text{Tr}K\right]$ is the minimal ring of definition of $K\text{.}$ This ring is equal to $\mathbb{Z}$ iff $K$ is the complexification of the Weyl group of an irreducible root system.

Proof is given in the Section 4.6.

It is easily seen from Table 1 and the theorem above that for the groups $K=\text{Lin}\hspace{0.17em}W,$ where $W$ is an infinite irreducible crystallographic $r\text{-group,}$ one has the following table:

Table 4
Linear parts of irreducible infinite crystallographic complex $r\text{-groups}$
$\mathbb{Z}\left[\text{Tr}K\right]$	$\mathbb{Z}$	$\mathbb{Z}\left[i\right]$	$\mathbb{Z}\left[2i\right]$	$\mathbb{Z}\left[i\sqrt{2}\right]$	$\mathbb{Z}\left[\omega \right]$	$\mathbb{Z}\left[2\omega \right]$	$\mathbb{Z}\left[\frac{1+i\sqrt{7}}{2}\right]$
$K$	$K_1=A_s,$ $s\ge 1\text{;}$ $G(2,1,s)=B_s,$ $s\ge 2\text{;}$ $G(2,2,s)=D_s,$ $s\ge 3\text{;}$ $G(6,6,2)=G_2\text{;}$ $K_{28}=F_4\text{;}$ $K_{35}=E_6\text{;}$ $K_{36}=E_7\text{;}$ $K_{37}=E_8\text{.}$	$G(4,1,s),$ $s\ge 2\text{;}$ $G(4,4,s),$ $s\ge 3\text{;}$ $G(4,2,s),$ $s\ge 3\text{;}$ $K_3$ $\text{(}m=4\text{);}$ $K_8\text{;}$ $K_{29}\text{;}$ $K_{31}\text{.}$	$G(4,2,2)$	$K_{12}$	$G(3,1,s),$ $s\ge 2\text{;}$ $G(6,1,s),$ $s\ge 2\text{;}$ $G(3,3,s),$ $s\ge 3\text{;}$ $G(6,6,s),$ $s\ge 3\text{;}$ $G(6,2,s),$ $s\ge 2\text{;}$ $G(6,3,s),$ $s\ge 3\text{;}$ $K_3$ $\text{(}m=3,6\text{);}$ $K_4\text{;}$ $K_5\text{;}$ $K_{25}\text{;}$ $K_{26}\text{;}$ $K_{32}\text{;}$ $K_{33}\text{;}$ $K_{34}\text{.}$	$G(6,3,2)$	$K_{24}$
fraction field of $\mathbb{Z}\left[\text{Tr}K\right]$	$\mathbb{Q}$	$\mathbb{Q}\left(\sqrt{-1}\right)$	$\mathbb{Q}\left(\sqrt{-1}\right)$	$\mathbb{Q}\left(\sqrt{-2}\right)$	$\mathbb{Q}\left(\sqrt{-3}\right)$	$\mathbb{Q}\left(\sqrt{-3}\right)$	$\mathbb{Q}\left(\sqrt{-7}\right)$

Further remarks

a) In contrast to the real case, there exist $1\text{-parameter}$ families of inequivalent irreducible complex infinite crystallographic $r\text{-groups}$ $W$ with a fixed linear part $\text{Lin}\hspace{0.17em}W$ (i.e. the groups with a fixed linear part may have moduli). We shall see below that an irreducible crystallographic $r\text{-group}$ $W$ with $\text{Lin}\hspace{0.17em}W=K$ has moduli iff $\mathbb{Z}\left[\text{Tr}K\right]=\mathbb{Z},$ i.e. iff $K$ is the complexification of the Weyl group of an irreducible root system.

b) It follows from Table 4 (and from a known result in algebraic number theory) that the ring $\mathbb{Z}\left[\text{Tr Lin}\hspace{0.17em}W\right],$ where $W$ is an infinite irreducible crystallographic $r\text{-group,}$ is always a unique factorisation domain. It would be interesting to have an a priori proof of this fact.

c) If $k=ℝ$ then it is known (and was a priori proved in 1948 - 51 by Cheval ley and Harish-Chandra) that there exists a bijective correspondence between the set of classes of equivalent infinite (hence crystallographic) $r\text{-groups}$ $\text{(}=$ affine Weyl groups) and the set of classes of isomorphic complex semisimple Lie algrebras.

Question: is it possible to attach to an infinite complex crystallographic $r\text{-group}$ a sort of "global object" (like a semisimple Lie algebra in the real case) in a such way that the correspondence between these $r\text{-groups}$ and "global objects" will be bijective? $$\begin{array}{ccc}\begin{array}{|c|}\hline \text{real}\\ \text{crystallographic}\\ r\text{-group}\\ \hline\end{array}& ⟷& \begin{array}{|c|}\hline \text{semisimple complex}\\ \text{Lie algebra}\\ \hline\end{array}\\ \begin{array}{|c|}\hline \text{complex}\\ \text{crystallographic}\\ r\text{-group}\\ \hline\end{array}& ⟷& \begin{array}{|c|}\hline \text{?}\\ \hline\end{array}\end{array}$$ We do not know whether such an object exists or not. It is funny that we can calculate (see 4.4) the group which, by analogy with the real case, might be "the center" of this hypothetical object.

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