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=.

Let W be an irreducible infinite r-group, WA(E). 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 (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). A linear subspace V of this real vector space is called a real form of V if

a) the natural map VV is an isomorphism, i.e. some (hence, any) -basis of V is a -basis of V;
b) the restriction ||V of | to V is realvalued (hence V is euclidean with respect to ||V).

If V is a real form of V then V is the complexification of V.

Let aE be a point. We can consider E as a real affine space of dimension 2n. The affine subspace E=a+V of this affine space is called a real form of E and E is called the complexification of E.

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(V) acts transitively on the set of real forms of V.
2) The group of motions of E acts transitively on the set of real forms of E.
3) Every motion γ of a euclidean affine space E can be extended in a unique way to a motion γ of E. This motion γ is called the complexification of γ (and γ is called the real form of γ).
4) dimHγ=dimHγ. Specifically, γ is a reflection iff γ is a reflection.

Proof.

Proof is left to the reader.

This proposition gives a method for constructing noncrystallographic infinite r-groups. Indeed, let GA(E) be an infinite (real) r-group. Then it is easy to see that G= {γ|γG} A(E) is an infinite complex noncrystallographic r-group (and G is irreducible if and only if G is).

Classification of infinite irreducible complex noncrystallographic r-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-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.

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 TranW. TranW=WTranA(E), cf. Section 1.1. It is clear that WW and W/TranWLinW. We usually identify TranW with a subgroup of the additive group of V by means of the map γvv. Clearly this subgroup is a LinW-invariant lattice in V.

It will be proven in Section 3.1 that TranW is a lattice of full rank (i.e. of rank 2n) and LinW is a finite group (hence, LinW is a finite irreducible complex linear r-group, see 1.4). Therefore, to describe W, one needs to point out a group LinWGL(V) from the Shephard and Todd list (i.e. from the theorem in 1.6), a LinW-invariant lattice TranWV of rank 2n and the way LinW and TranW are "glued" together. This is done below as follows:

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

2) TranW is described explicitly by linear combinations of vectors ej, 1js, that generate TranW. Here Rj=Rej,θj, 1js, is a fixed generating system of reflections of LinW which is related to the graph of LinW given in 1) as described in 1.6. To point out the vectors ej, 1js, explicitly, we assume that V is a subspace of a standard hermitian infinitedimensional coordinate space i.e. the space, whose elements are the sequences (a1,a2,) with only a finite number of nonzero elements aj, and a scalar product defined by the formula (a1,a2,)|(b1,b2,) =j=1ajbj. The vectors ej, 1js, are given by their, coordinates on a standard basis ε1,ε2, of , where εj= (0,,0,1,0,).

3) The problem how to describe the "glueing" of LinW and TranW comes down to the determination of an extension of TranW by LinW, 0TranWWLinW1. Therefore it is done by means of cohomology. Let us show how it can be done.

Cohomology

Let G be a subgroup of A(E) and write T=TranG, K=LinG. Choose a point aE. Take PK and let γG be such that Linγ=P. We have κa(γ)= (P,s(P)), s(P)V. It is easy to see that the map s:KV/T, s(P) =s(P)+T, is well defined and is in fact a 1-cocycle, i.e. s(PQ) =s(P) +Ps(Q) ,P,QK. (here K acts on V/T in the natural way).

Vice versa, if r:KV/T is an arbitrary 1-cocycle, let us consider an arbitrary map r:KV such that r(P)= r(P)+T,PK. Then the set { (P,r(P)+t) |tT,PK } is a subgroup H of A(E) with LinH=K and TranH=T.

If we replace a by an other point bE, then (see 1.1). κb(γ)=κa (γa-bγγb-a) = ( P,s(P)+ v-Pv 1-coboundary ) ,wherev=a-b. Therefore we have a bijection between the set of TranA(E)-conjugacy classes of subgroups G of A(E) with LinG=K, TranG=T and the group H1(K,V/T).

However we have to consider subgroups of A(E) up to equivalence, i.e. up to A(E)-conjugation (and not just up to TranA(E)-conjugation)! This can be done as follows by means of an extra relation on H1(K,V/T). Let N(K,T)= { QGL(V) |QKQ-1 =K,QT=T } . If QN(K,T) and s:KV/T is a 1-cocycle, resp. 1-coboundary, then it is easy to check that the map Q(s):K V/T given by the formula Q(s)(P) =Qs (Q-1PQ), PK, is again a 1-cocycle, resp. 1-coboundary (here Q acts on V/T in the natural way). Therefore we have an action of N(K,T) on H1(K,V/T) (clearly, by means of automorphisms).

Let δA(E) be such that LinδGδ-1=K, TranδGδ-1=T. We want to calculate the cocycle that corresponds to δGδ-1. Changing δ to δγv, where v=(Linδ-1)(a-δ(a)), we can assume that κa(δ)=(Q,0), QN(K,T). Let PK and λG be such that κa(λ)=(Q-1PQ,s(Q-1PQ)). Then κa(δλδ-1)=(P,Qs(Q-1PQ)). Therefore the cocycle corresponding to δGδ-1 is Q(s) where s is the cocycle corresponding to G.

We see now that there is a bijection between the set of classes of equivalent subgroups GA(E) with LinG=K, TranG=T and the set of N(K,T)-orbits in H1(K,V/T).

With all these facts in mind, we determine the extension W (of TranW by LinW) by pointing out a 1-cocycle which represents the corresponding element of H1(LinW,V/TranW) (in fact, the whole N(LinW,TranW)-orbit in H1(LinW,V/TranW)). In order to do so, we need only give the values of this 1-cocycle on the elements of a generating system of reflections of LinW. Technically it is more convenient to realize it as follows.

Let LinW˜ be a free group with generators rj, 1js. We have an epimorphism ϕ:LinW˜LinW, ϕ(rj)=Rj, 1js. The kernel of ϕ is the subgroup of "relations" of LinW. This epimorphism leads in a natural way to an action of LinW˜ on V. A 1-cocycle c of LinW˜ with values in V is given by its values on the generators rj, c(rj),1js, and these values may be arbitrary (because LinW˜ is free). It is easy to see that the formula Rjc(rj)+ TranW,1js, defines a 1-cocycle of LinW with values in V/TranW iff c(F)TranW for every FKerϕ. It is also clear that every 1-cocycle of LinW with values in V/TranW is obtained in such a way.

We shall give the extension W (of TranW by LinW) by writing down the vectors c(rj), 1js.

We are now ready to formulate the results of the classification of infinite irreducible crystallographic r-groups.

Denote by Kb the finite linear irreducible r-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 K5,K6).

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 KGL(V) be an irreducible finite r-group. Then the following properties are equivalent:

a) There exists a nonzero K-invariant lattice in V.
b) There exists a K-invariant lattice of rank 2n in V.
c) K=LinW where W is an infinite crystallographic r-group.
d) The ring with unity, generated over by all cyclic products of a graph of K, lies in the ring of algebraic integers of a purely imaginary quadratic extension of .
e) K is defined over a purely imaginary quadratic extension of .
f) K is one of the groups: K1; K2(m=2,3,4,6); K3(m=2,3,4,6); K4; K5; K8; K12; K24; K25; K26; K28; K29; K31; K32; K33; K34; K35; K36; K37.

Proof.

Proof is given in Section 4.6.

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: Ω={z|-12Rez<12,|z|1 if Rez0 and |z|>1 if Rez>0} - this is the "modular strip"; [α,β]={aα+bβ|a,b} for arbitrary α,β).

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

The proof is given in the subsequent chapters.

Table 2
The irreducible infinite crystallographic complex r-groups.
Notation of W n=dimW LinW TranW e1,,es cocycle c
[As]α
s1
s K1,
type As
s1
[1,α]e1++[1,α]es,
αΩ
ej=(εj-εj+1)/2
j=1,,s
c=0
[G(2,1,s)]1α
s3
s K2

type
G(2,1,s)
s3
[1,α]e1+ [1,α]2e2++ [1,α]2es,
αΩ
e1=ε1,
ej=(εj-1-εj)/2
j=2,,s
[G(2,1,s)]2β
s3
[1,β]e1+ [1,1+β2]2e2++ [1,1+β2]2es,
βΩ
[G(2,1,s)]3γ
s3
[1,γ]e1+ [12,γ]2e2++ [12,γ]2es,
γΩ
[G(2,1,s)]4δ
s3
[1,γ]e1+ [1,δ2]2e2++ [<1,δ2]2es,
δΩ
[G(2,1,s)]5λ
s3
[1,λ]e1+ [12,λ2]2e2++ [12,λ2]2es,
λΩ
[G(3,1,s)]1
s2
s K2
type
G(3,1,s)
s2
[1,ω]e1+ [1,ω]2e2++ [1,ω]2es
[G(3,1,s)]2
s2
[1,ω]e1+ [1,ω]i23e2++ [1,ω]i23es
[G(4,1,s)]1
s2
s K2
type
G(4,1,s)
s2
[1,i]e1+ [1,i]2e2++ [1,i]2es
[G(4,1,s)]2
s2
[1,i]e1+ [1,i]εe2++ [1,i]εes
[G(6,1,s)]
s2
s K2
type
G(6,1,s)
s2
[1,ω]e1+ [1,ω]2e2++ [1,ω]2es
[G(2,2,s)]α
s3
s K2
G(2,2,s)
s3
[1,α]e1++[1,α]es,
αΩ
e1=-(ε1+ε2)/2,
ej=(εj-1-εj)/2,
j=2,,s
[G(3,3,s)]
s3
s K2,
type
G(3,3,s)
s3
[1,ω]e1++[1,ω]es e1=ωε1-ε2,
ej=(εj-1-εj)/2,
j=2,,s
[G(4,4,s)]
s3
s K2
type
G(4,4,s)
s3
[1,i]e1++[1,i]es e1=(iε1-ε2)/2,
ej=(εj-1-εj)/2,
j=2,,s
[G(6,6,s)]
s3
s K2,
type
G(4,4,s)
s3
[1,ω]e1++[1,ω]es e1=((1+ω)e1-e2)/2
ej=(εj-1εj)/2
j=2,,s
[G(2,1,2)]1α 2 K2,
type
G(2,1,2)
= type
G(4,4,2)
[1,α]e1+[1,α]2e2,
αΩ
e1=ε1
e2=(ε1-ε2)/2
[G(2,1,2)]2β [1,β]e1+[1,β2]2e2,
βΩ
[G(2,1,2)]3γ [1,γ]e1+[1,1+γ2]2e2,
γΩ
[G(6,6,2)]1α 2 K2,
type
G(6,6,2)
[1,α]e1+[1,α](2+ω)e2,
αΩ
e1=((1+ω)ε1-ε2)/2
e2=(ε1-ε2)/2
[G(6,6,2)]2β [1,β]e1+[1,β3](2+ω)e2,
βΩ
[G(6,6,2)]3γ [1,γ]e1+[1,1+γ3](2+ω)e2,
γΩ
[G(6,6,2)]4δ [1,δ]e1+[1,2+δ3](2+ω)e2,
δΩ
[G(4,2,s-1)]1
s3
s-1 K2, type
G(4,2,s-1)
s3
T=[1,i]e1++[1,i]es-1 e1=(iε1-ε2)/2
ej=(εj-1εj)/2
j=2,,s-1
es=εs-1
[G(4,2,s-1)]1*
s3
c(rj)=0,
j=1,,s-1
c(rs)=es/2
[G(4,2,s-1)]2
s3
T(T+1+i2(e1+e2))=
[1,i]e1++[1,i]es-1+12[1,i]es
c=0
[G(4,2,2)]3 2 K2,
type
G(4,2,2)
[1,i]e1+[1,i](1+i)e2
[G(6,2,s-1)]1
s3
s-1 K2
type
G(6,2,s-1)
s3
[1,ω]e1++[1,ω]es-1 e1=((1+ω)ε1-ε2)/2,
ej=(εj-1-εj)/2,
j=2,,s-1
es=εs-1
[G(6,2,2)]2 2 K2,
type
G(6,2,2)
[1,ω]e1+[1,ω](2+ω)e2
[G(6,3,s-1)]1
s3
s-1 K2, type
G(6,3,s-1)
s3
[1,ω]e1++[1,ω]es-1
[G(6,3,2)]2 2 K2,
type
G(6,3,2)
[1,2ω]e1+[2,ω]e2
[K3(3)] 1 K3
m=3
[1,ω]e1 e1=ε1
[K3(4)] K3
m=4
[1,i]e1
[K3(6)] K3,
m=6
[1,ω]e1
[K4] 2 K4 [1,ω]e1+[1,ω]e2 e1=ε1,
e2=1-ω3(ε1+ε2+ε3)
[K5] K5 [1,ω]e1+[1,ω]2e2 e1=ε1,
e2=1-ω3(2ε1+ε2)
[K8] K8 [1,i]e1+[1,i]e2 e1=ε1,
e2=1-i2(ε1-ε2)
[K12] K12 [1,i2]e1+[1,i2]e2 e1=12ε1+1+i2ε2,
e2=2+(2-2)i4ε1+2+2-2i4ε2,
e3=12ε1+1-i2ε2
[K12]* c(r1)=c(r2)=0
c(r3)=1+i2e3
[K24] 3 K24 [1,1+i72]e1+ [1,1+i72]e2+ [1,1+i72]e3 e1=ε2,
e2=(1-i7)(ε2+ε3)/4,
e3=(-ε1-ε2+1+i72ε3)/2
c=0
[K25] K25 [1,ω]e1+[1,ω]e2+[1,ω]e3 e1=ε3,
e2=1-ω3(ε1+ε2+ε3),
e3=-ωε2
[K26]1 K26 [1,ω]e1+[1,ω]e2+[1,ω]2e3 e1=1-ω23(ε1+ε2+ε3),
e2=ε3,
e3=12(ε2-ε3)
[K26]2 [1,ω]e1+[1,ω]e2+[1,ω]o23e3
[F4]1α 4 K28

F4
[1,α]e1+[1,α]e2+[1,α]2e3+[1,α]2e4,
αΩ
ej=(εj+1-εj+2)/2,
j=1,2,
e3=ε4,
e4=(ε1-ε2-ε3-ε4)/2
[F4]2β [1,β]e1+[1,β2]e2+[1,β]2e3+[1,β2]2e4,
βΩ
[F4]3γ [1,γ]e1+[1,γ]e2+[1,1+γ2]2e3+
[1,1+γ2]2e4, γΩ
[K29] K29 [1,i]e1+[1,i]e2+[1,i]e3+[1,i]e4 e1=12(ε2-ε4)
e2=12(-iε2+ε3)
e3=12(-ε3+ε4)
e4=-1+i22(ε1+ε2+ε3+ε4)
[K31] K31 [1,i]e1+[1,i]e2+[1,i]e3+[1,i]e4 e1=12(ε2-ε4)
e2=12(-iε2+ε3)
e3=12(-ε3+ε4)
e4=-1+i22(ε1+ε2+ε3+ε4)
e5=1-i2ε4
[K31]* c(rj)=0
j=1,2,3,4,
c(r5)=1+i2e5
[K32] K32 [1,ω]e1+[1,ω]e2+[1,ω]e3+[1,ω]e4 e1=ε3,
e2=1-ω3(ε1+ε2+ε3),
e3=-ωε2,
e4=ω2-ω3(-ε1+ε2+ε4)
c=0
[K33] 5 K33 [1,ω]e1++[1,ω]en e1=ω2(ε5+ε6),
e2=-ω22(-ε1+(1+2ω)ε2
+ε3+ε4+ε5+ε6),
ej=12(εj-2-εj-1),
j=3,4,,n
[K23] 6 K24
[E6]α K35,
E6
[1,α]e1++[1,α]en,
αΩ
e1=(ε1-ε2-ε3-ε4
-ε5-ε6-ε7+ε8)/22,
e2=(ε1+ε2)/2,
ej=(-εj-2+εj-1)/2,
j=3,,n.
[E7]α 7 K36,
E7
[E8]α 8 K37,
E8

Equivalence

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

Table 3
Pairs of equivalent irreducible infinite crystallographic complex r-groups
W W condition
[G(2,1,s)]21+ω, s3 [G(2,1,s)]31+ω, s3 --
[G(2,1,s)]21+ω, s3 [G(2,1,s)]41+ω, s3 --
[G(2,1,s)]3i, s3 [G(2,1,s)]4i, s3 --
[G(2,1,2)]2β [G(2,1,2)]2-2/β -2/βΩ
[G(2,1,2)]21+ω [G(2,1,2)]31+ω --
[G(2,1,2)]2β [G(2,1,2)]31-2/β 1-2/βΩ
[G(2,1,2)]2γ [G(2,1,2)]3(γ-1)/(γ+1) (γ-1)/(γ+1)Ω
[G(2,1,2)]2β [G(2,1,2)]3-1-2/β -1-2/βΩ
[G(6,6,2)]2β [G(6,6,2)]2-3/β -3/βΩ
[G(6,6,2)]3γ [G(6,6,2)](2γ-1)/(γ+1) (2γ-1)/(γ+1)Ω
[G(6,6,2)]2β [G(6,6,2)]-1+3/β -1+3/βΩ
[G(6,6,2)]2β [G(6,6,2)]32-3/β 2-3/βΩ
[F4]2β [F4]2-2/β -2/βΩ
[F4]21+ω [F4]31+ω --
[F4]2β [F4]31-2/β 1-2/βΩ
[F4]3γ [F4]3(γ-1)/(γ+1) (γ-1)/(γ+1)Ω
[F4]2β [F4]3-1-2/β -1-2/βΩ

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

The structure of an extension of TranW by LinW.

As we have seen in Section 1.5, if k= then the structure of an infinite irreducible r-group W as an extension of TranW by LinW is very simple: it is always a semidirect product. The situation is more complicated when k=, because there exist infinite irreducible complex crystallographic r-groups W which are not semidirect products of TranW and LinW.

Theorem. The groups W from Table 2 which are not semidirect products of TranW and LinW are [G(4,2,s)]1*, [K12]*and [K31]*.

Theorem. Let KGL(V) be a finite irreducible r-group and let TV be a K-invariant lattice. Assume that there exists a crystallographic r-group W with LinW=K, TranW=T. Then the set of those elements of H1(K,V/T) which correspond to such subgroups W is in fact a subgroup of H1(K,V/T) and the order of this subgroup is 2.

The rings and fields of definition of LinW.

As we have seen in Section 1.5, if k= then the group LinW for an infinite irreducible r-group W is defined over . If k= then LinW for an infinite irreducible crystallographic r-group W is defined over a certain purely imaginary quadratic extension of , see the theorem in Section 2.5. We can describe this extension precisely.

Theorem. Let KGL(V) be a finite irreducible complex r-group. Then the ring with unity generated over by the set of all cyclic products related to an arbitrary fixed generating system of reflections of K coincides with the ring [TrK] generated over by the set of traces of all elements of K. The ring [TrK] is the minimal ring of definition of K. This ring is equal to 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=LinW, where W is an infinite irreducible crystallographic r-group, one has the following table:

Table 4
Linear parts of irreducible infinite crystallographic complex r-groups
[TrK] [i] [2i] [i2] [ω] [2ω] [1+i72]
K K1=As, s1;
G(2,1,s)=Bs, s2;
G(2,2,s)=Ds, s3;
G(6,6,2)=G2;
K28=F4;
K35=E6;
K36=E7;
K37=E8.
G(4,1,s), s2;
G(4,4,s), s3;
G(4,2,s), s3;
K3 (m=4);
K8;
K29;
K31.
G(4,2,2) K12 G(3,1,s), s2;
G(6,1,s), s2;
G(3,3,s), s3;
G(6,6,s), s3;
G(6,2,s), s2;
G(6,3,s), s3;
K3 (m=3,6);
K4;
K5;
K25;
K26;
K32;
K33;
K34.
G(6,3,2) K24
fraction
field of
[TrK]
(-1) (-1) (-2) (-3) (-3) (-7)

Further remarks

a) In contrast to the real case, there exist 1-parameter families of inequivalent irreducible complex infinite crystallographic r-groups W with a fixed linear part LinW (i.e. the groups with a fixed linear part may have moduli). We shall see below that an irreducible crystallographic r-group W with LinW=K has moduli iff [TrK]=, 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 [Tr LinW], where W is an infinite irreducible crystallographic r-group, is always a unique factorisation domain. It would be interesting to have an a priori proof of this fact.

c) If k= 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-groups (= 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-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-groups and "global objects" will be bijective? realcrystallographicr-group semisimple complexLie algebra complexcrystallographicr-group ? 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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