Discrete complex reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 or .


Let E be an affine space over k, dimE=n, and V be its space of translations. If vV, we denote by γv the corresponding translation of E, i.e. γv(a)=a+v, aE. Let A(E) be the group of all affine transformations of E and TranA(E)= {γv|vV}. We denote by GL(V)·V the natural semidirect product of GL(V) and V. Its elements are pairs (P,v), where PGL(V), vV, and the group operations are given by formulas (P,v)(Q,w) = (PQ,Pw+v), (P,v)-1 = (P-1,-P-1v). Let Lin:A(E) GL(V) be the standard homomorphism defined by formula γ(a+v)=γ(a)+ (Linγ)v, γA(E),aE, vV. If we take a point aE as origin, we obtain an isomorphism κa:A(E) GL(V)·V given by the formula κa(γ)= (Linγ,γ(a)-a). Identifying A(E) and GL(V)·V by means of κa, we obtain the action of GL(V)·V on E given by the formula (P,v)q=a+P (q-a)+v,qE. The dependence on a is given by κb(γ)=κa (γa-bγγb-a), γA(E), bE. For every γA(E) and PGL(V) we use the notations Hγ = {aE|γ(a)=a}, HP = {vV|Pv=v}. These are subspaces of E and V respectively.

Let | be a positive definite inner product on V, i.e. V is an euclidian (k=), resp. hermitian (k=) linear space with respect to | (linear in the first coordinate). Let also U(V)={PGL(V)|Ppreserves|}; 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 ρ(a,b)= a-b|a-b, a,bE.

Motions and reflections

We say that γA(E) is a motion of E if γ preserves the distance ρ. It is easy to see that γ is a motion iff LinγU(V).

Definition. An affine reflection γA(E) is an element with the properties:

1) γ is a motion,
2) γ has finite order,
3) codimHγ=1.
A linear reflection RGL(V) is an element with the properties:
1) RU(V),
2) R has finite order,
3) codimHR=1.
The subspaces Hγ and HR are called the mirrors of γ, resp. R.

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 R={vV|vHR} is called the root line of R. If vR, v|v=1, then Rv=θv, where 01 is a primitive root of 1 (if k= then θ=-1, if k= then θ may be arbitrary). The pair (v,θ) determines R completely and every pair (u,η), with u|u=1, η1 a primitive root of unity (=-1 if k=), can be obtained in such a way from a reflection. We shall write R=Rv,θ Some properties of the reflections are contained in the following

Proposition. Let γA(E), aE and κa(γ)=(R,v). Then

1) γ is a reflection iff R is a reflection and vHR.
2) If γ is a reflection and R=Re,θ then Hγ=a+HR+ (1-θ)-1v.
3) Re,θv=v-(1-θ)v|ee.
4) If γ is a reflection and δ is a motion then δγδ-1 is a reflection.


Proof is left to the reader.

Main problem

We shall say that a subgroup W of A(E) is an r-group if it is discrete and generated by affine reflections.

If E and E are two affine spaces and GA(E) and GA(E) are two arbitrary subgroups, then we shall say that G and G are equivalent if there exists an affine bijection ϕ:EE such that W=ϕWϕ-1.

This means that after identifying E and E by means of an arbitrary fixed isomorphism, the groups G and G, as subgroups of A(E), have to be conjugate in A(E).

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

Our main concern in these lectures will be to classify r-groups up to equivalence.

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


Let W be a subgroup of A(E). We shall say that W is reducible if there exist affine metric spaces Ej, j=1,,m,m2, and subgroups Wj of A(Ej) such that W is equivalent to W1××WmA(E1××Em). 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 WA(E) be a nontrivial subgroup generated by affine reflections (possibly not discrete). Then:

a) W is equivalent to a product W1××Wm where Wj, 1jm, are irreducible groups, and Wj is either generated by affine reflections or trivial (hence 1-dimensional), but not all Wj are trivial.
b) W is irreducible iff LinW is an irreducible linear group (generated by linear reflections).
c) Wj, 1jm, are uniquely defined up to equivalence and order.
d) Every product of the type described in a) is a group generated by reflections.


a) The statement follows from the equality H(γ1,,γm) =Hγ1××Hγm (hence (γ1,,γm) is a reflection iff one and only one of γj, 1jm, 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 LinW lies in U(V). Therefore, LinW is a completely reducible linear group. Let V=j=1m Vj where Vj, 1jm, are irreducible LinW-modules. Consider the subspaces Ej=a+Vj, 1jm, where aE is an origin, and let πj:WA(Ej), 1jm, be the morphism given by the formula πj(γ)=κa-1 ( Linγ|Vj, pj(γ(a)-a) ) (here pj:VVj is the natural projection). Let Wj=πj(W). Then it is not difficult to check that the map ϕ:EE1××Em, given by the formula ϕ(q)= ( a+p1(q-a), ,a+pm(q-a) ,qE, ) defines an equivalence of W and W1××Wm.

c) Suppose that WA(E) and WA(E) are two equivalent groups generated by affine reflections and let ϕ:EE establish the equivalence of these groups. Let W=W1××Wr and W=W1××Ws be decompositions into products of irreducible groups and let E=E1××Er, E=E1××Es, V=V1Vr, V=V1Vs, be the corresponding decompositions of the affine spaces and its spaces of translations. We consider Wj and W1 as subgroups of W and W resp.

It is clear that Linψ yields the equivalence of the linear groups LinW and LinW (in the usual sense). Hence (Linψ)Vj is a simple LinW-submodule of V for every j.

Let γW be a reflection. Then γWp for some p, see a) above. But ψγψ-1=γ is also a reflection (inside W). Therefore γWq for some q.

The root line of Linγ is contained in Vq and this line is (Linψ), where is a root line of Linγ. Clearly, Vp. Hence Vq (Linψ)Vp 0. It follows now from the irreducibility that in fact Vq= (Linγ)Vp. Therefore ψVpψ-1Wq (because Wp is generated by all reflections τ for which Linτ has its root line in Vp). The same proof is valid for the inverse inclusion. Hence, ψWpψ-1=Wq and we can proceed by induction.

d) This is clear (see the equality in a)).

Therefore we shall from now on only consider the case of irreducible r-groups.

What is already known: case k=.

Let W be an irreducible r-group in A(E).

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. Indeed, let aE be an arbitrary point. Then b=a+1|W| γW (γ(a)-a) is fixed under W. Therefore κb gives an isomorphism of W with LinW, 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=.

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 H3 or H4.

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 Rj, 1jn, which are reflections in the faces of a Weyl chamber C. There exists a unique set of vectors ej, 1jn, of unit length with the property: Rj=Rej,θj (where in fact θj=-1) and C={vV|ej|v>0,1jn}. The angle between the mirrors HRi and HRj is of the form πmij, mij; mij2. The nodes of the Coxeter graph of W are in bijective correspondence with the reflections Rj, 1jn. Two nodes Ri and Rj are connected by an edge iff mij3. The weight of this edge is equal to mij (if mij=3 then the weight is usually omitted). The complete list of finite irreducible r-groups is given by the following table of Coxeter graphs. An, 4 F4, 4 Bn=Cn, 6 G2, Dn, 5 H3, E6, 5 H4, E7, p I2(p),p=5orp7. E8, Let us consider the numbers cij= (1-θi) (1-θj) ei|ej ej|ei =4cos2πmij. One can change the weight mij to the numbers cij 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 LinW, 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. The groups LinW, thus obtained are distinguished from the others of the list above in the following way (see [Bou1968], Ch. VI):

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

a) K=LinW where W is an infinite r-group.
b) There exists a K-invariant lattice in V of rank n.
c) K is defined over .
d) K is the Weyl group of a certain irreducible root system, i.e. a group whose Coxeter graph is one of An, Bn, Dn, E6, E7, E8, F4, G2.
e) All the numbers cij lie in .
f) The ring with unity, generated over by all of the numbers cij, coincides with .

What is already known: case k=.

Let k= and WA(E) be an irreducible r-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 Rj=Rej,θj, 1js, be a generating system of reflections of W. We can assume that θj=e2πimj. Therefore this system (and hence W) is uniquely defined by the system of lines Rj in V with the multiplicities mj, 1js.

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 {j}jJ and {j!}jJ be two sets of lines in V and let ejj, ej!j! be arbitrary vectors with 1=ej|ej=ej!|ej!. For any finite set of indices j1,,jmJ let us consider the numbers hj1jm= ej1|ej2 ej2|ej3 ejm-1|ejm ejm|ej1 and hj1jm!= ej1!|ej2! ej2!|ej3! ejm-1!|ejm! ejm!|ej1!. Then hj1jm, resp. hj1jm!, is independent of the choice of the vectors ej, resp. ej!, jJ. Moreover, the systems {j}jJ and {j}jJ are isometric (i.e. gj=j, jJ, for a certain gU(V)) if and only if hj1jm= hj1jm! (*) for arbitrary j1,,jmJ.


We need only prove that if (*) is fulfilled then for every iJ there exists a number λi such that ej!|ek!= λjej|λkek for every j,kJ, i.e. the systems {ej!}jJ and {λjej}jJ have the same Gram matrix (all the other statements are evident).

Let us fix an index tJ. It is not difficult to see that one can assume that the system {j}jJ is "connected", i.e. there exists for every jJ a sequence j1,,jmJ such that j1=j, jm=t and ejs|ejs+10, s=1,,m-1. It is now a matter of straightforward computation to check that one can take λj= s=1m-1 ejs+1!|ejs! ejs+1|ejs .

We have seen above that an r-group is defined by a system {j,mj}jJ of lines jV with the multiplicities mj. It follows from the proposition that such a system is uniquely (up to isometry) defined by a system of numbers cj1jm= hj1jm s=1m (1-e2πi/mjs) (one can derive from these numbers the multiplicities because cj=1-e2πi/mj). It will become clear later on why hj1jm is multiplied by s=1m(1-e2πi/mjs) (and not, say by s=1me2πi/mjs); the numbers cj1jm 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 cj1jm, i.e. those with all indices j1,,jm distinct, because cl1lp-1llp+1lq-1llq+1lr= cl1lp-1llq+1lr· cllp+1lq-1· l l1 l2 lp-1 lp-1 lp+1 lq+1 lp

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

a) If j1,,jm is a cyclic permutation of j1,,jm then cj1jm= cj1jm. In other words, cj1jm depends only on a cycle σ=(j1,,jm). We use therefore the notation cσ=cj1jm. In particular, cjk=ckj. b) If σ=(j1,,jm) then cσcσ-1= cj1j2 cj2j3 cjm-1jm cjmj1 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: cl1l2lpjsjs-1j1· clplp+1lq-1lql1j1j2js =cl1l2lp-1lplp+1lq-1lq· cl1j1·cj1j2 cjs-1js· cjslp lq-1 lq l1 l2 l3 j1 j2 js-1 js lp+2 lp+1 lp lp-1 lp-2

A system of lines jV with the multiplicities mj, jJ, can be described by a graph as follows.

The nodes of the graph are in bijective correspondence with the lines j, jJ. If a node represents the line j then this node has the weight mj. Two nodes j and k are connected by an edge iff cjk0 and if they are connected then the weight of the edge is equal to cjk. It will be convenient to omit the weight of the node, resp. the edge, if it is equal to 2, resp. 1. Moreover, every simple cycle of this graph is supplied with an arbitrary (but fixed) orientation and has weight equal to the corresponding cycle product: cj1j2js mj2 mj1 mjs mjk-1 mjk mjk+1 cj1j2 cj1js cjk-1jk cjkjk+1

Therefore we now have a way to represent a finite r-group WGL(V) with a fixed generating system of reflections by means of a graph corresponding to the system of lines RjV with the multiciplicities mj, 1js. (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-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-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: Notation: ω=e2πi/3 η=e2πi/5 ζm=e2πi/m ε=e2πi/8

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 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-groups). Any irreducible complex finite r-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-groups.
No Type Graph Ring generated by cyclic products dimV
1 As,
1 s s
2 G(m,1,s),
m2, s2,
type Bs=Cs
if m=2
1 2 s 1-ζm m
1 2 s 2 if m=2
[ω] if m=3,6,
[i] if m=4,
if m=2
2 G(m,m,s),
m2, s3,
type Ds
if m=2
1 2 3 s 4cos2πm 2ζ2mcosπm ζ2m=eπi/m [e2πi/m],
[ω] if m=3,6,
[i] if m=4,
1 2 3 s
if m=2
if m=2
2 G(m,m,2)=I2(m),
A2 if m=3,
B2 if m=4,
G2 if m=6
1 2 4cos2πm
1 2 if m=3,
1 2 2 if m=4,
1 2 6 if m=6
if m=3,4,6
2 G(m,p,s-1),
m2, s4,
1 2 3 s-1 s m/p 4cos2πm 2ζ2mcosπm 1-ζmp [e2πi/m],
[i] if m=4, p=2,
[ω] if m=6, p=2 and m=6, p=3
2 G(m,p,2),
1 2 3 m/p 4cos2πm 2(1-ζmp)ζ2mcosπm 1-ζmp 1-ζmp [ζm,2cosπm,ζm(1-ζmp)],
[2i] if m=4, p=2,
[ω] if m=6, p=2,
[2ω] if m=6, p=3
3 []m 1 m [e2πi/m],
[ω] if m=6,3,
[i] if m=4,
if m=2
4 3[3]3 1 2 -ω 3 3 [ω] 2
5 3[4]3 -2ω 3 3 [ω] 2
6 3[6]2 1 2 iω2+1-ω 3 [e2πi/12] 2
7 3,3,26 1 2 3 iω2+1-ω -2(i+ω) iω2+1-ω -2ω 3 3 [e2πi/12] 2
8 4[3]4 1 2 -i 4 4 [i] 2
9 4[6]2 1 2 1-i-ε3 4 [ε] 2
10 4[4]3 1 2 -i-ω 1 4 3 3 [e2πi/12] 2
11 4,3,212 1 2 3 iε-i+1 -2iεω2-ε-i+ 2iω2-ω 2iω2-ω+1 -i-ω 4 3 [e2πi/24] 2
12 GL(2,3) 1 2 3 2 4+i2 3 3 [i2] 2
13 4,3,22 1 2 3 2 2+3+i 3 2+2 [i,2] 2
14 3[8]2 1 2 iω22+1-ω 3 3 [ω,i2] 2
15 4,3,26 1 2 3 iω2+1-ω (2+1)iω2+1-ω 2 2iω2+1-ω 3 [i,ω,2] 2
16 5[3]5 1 2 -η 5 5 [η] 2
17 5[6]2 1 2 iη3+1-η 5 [e2πi/20] 2
18 5[4]3 1 2 -ω-η 5 3 [e2πi/15] 2
19 5,3,230 1 2 3 iη3+1-η iω2η3+iη3-η -iω2η2-ω iη2ω2(1+η)+1-ω -ω-η 5 3 [e2πi/60] 2
20 3[5]3 1 2 -ω(η+η4+2) 3 3 [ω,1+52] 2
21 3[10]2 1 2 iω2(η+η4+1)-ω+1 3 [ω,i1+52] 2
22 5,3,22 1 2 3 2 (i+1)(η+η4)+4 η+η4+3 3 [5-12,i5-12] 2
23 H3 1 2 3 3+52 [1+52] 3
24 J3(4) 1 2 3 2 1+i72 [1+i72] 3
25 L3 1 2 3 -ω -ω 3 3 3 [ω] 3
26 M3 1 2 3 -ω 1-ω 3 3 [ω] 3
27 J3(5) 1 2 3 -2ωcosπ2 4cos2π5 [ω,1+52] 3
28 F4 1 2 3 4 2 4
29 [21;1]4=N4 1 2 3 4 i [i] 4
30 H4 1 2 3 4 3+52 [1+52] 4
31 [(12γ34)+1]=EN4 1 2 2 2 2 3 4 5 i i-1 [i] 4
32 L4 1 2 3 4 -ω -ω -ω 3 3 3 3 [ω] 4
33 [21;2]3=K5 1 2 3 4 5 1+ω [ω] 5
34 [21;3]3=K6 1 2 3 4 5 6 1+ω [ω] 6
35 E6 1 3 4 5 6 2 6
36 E7 1 3 4 5 6 7 2 7
37 E8 1 3 4 5 6 7 8 2 8

Remark. For those groups among them that are of the form LinW, where W is an infinite irreducible complex r-group, one can obtain an explicit expression for the set of lines rj, 1js, from Table 2 below (by taking a vector ej of unit length in each Rr, 1js). 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.


We consider the case where n=dimE=1.

Let aE be a point. We identify A(E) and GL(V)·V by means of κa, see Section 1.1.

Let Γ0 be a lattice in V. (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={(±1,t)|tΓ} of GL(V)·V. This subgroup acts on E (see Section 1.1) and it is easy to check that it is an infinite irreducible complex r-group.

There are two possibilities: either rkΓ=1 or rkΓ=2. If rkΓ=1 i 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-group on Γ.

If rkΓ=2 0 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-group to be noncrystallographic, see Section 1.5.

Denote by Λ a lattice of· equilateral triangles in V (=), 0 and by L a lattice of squares in V (=) 0 Then it is not difficult to check that the groups {(ωl,t)|tΛ,l}, {(±ωl,t)|tΛ,l}, and {(il,t)|tL,l} are infinite irreducible complex crystallographic r-groups.

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

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