Last update: 12 May 2014
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.
We assume in this chapter that the ground field be or
Let be an affine space over and be its space of translations. If we denote by the corresponding translation of i.e. Let be the group of all affine transformations of and We denote by the natural semidirect product of and Its elements are pairs where and the group operations are given by formulas Let be the standard homomorphism defined by formula If we take a point as origin, we obtain an isomorphism given by the formula Identifying and by means of we obtain the action of on given by the formula The dependence on is given by For every and we use the notations These are subspaces of and respectively.
Let be a positive definite inner product on i.e. is an euclidian resp. hermitian linear space with respect to (linear in the first coordinate). Let also this is a compact group. The space becomes a euclidean, resp. hermitian affine metric space with respect to the distance given by the formula
We say that is a motion of if preserves the distance It is easy to see that is a motion iff
Definition. An affine reflection is an element with the properties:
|1)||is a motion,|
|2)||has finite order,|
|2)||has finite order,|
Sometimes we s!1all simply say reflection when it is clear what we are talking about.
If is a linear reflection then the line is called the root line of If then where is a primitive root of (if then if then may be arbitrary). The pair determines completely and every pair with a primitive root of unity if can be obtained in such a way from a reflection. We shall write Some properties of the reflections are contained in the following
Proposition. Let and Then
|2)||If is a reflection and then|
|4)||If is a reflection and is a motion then is a reflection.|
Proof is left to the reader.
We shall say that a subgroup of is an if it is discrete and generated by affine reflections.
If and are two affine spaces and and are two arbitrary subgroups, then we shall say that and are equivalent if there exists an affine bijection such that
This means that after identifying and by means of an arbitrary fixed isomorphism, the groups and as subgroups of have to be conjugate in
We want to emphasize here that even when and are affine metric spaces, need not to be distance preserving.
Our main concern in these lectures will be to classify up to equivalence.
We shall show now that in solving this problem one can restrict attention to irreducible groups.
Let be a subgroup of We shall say that is reducible if there exist affine metric spaces and subgroups of such that is equivalent to Otherwise is called irreducible. Clearly, every group is equivalent to a product of irreducible groups (but its decomposition need not be unique).
Theorem. Let be a nontrivial subgroup generated by affine reflections (possibly not discrete). Then:
|a)||is equivalent to a product where are irreducible groups, and is either generated by affine reflections or trivial (hence but not all are trivial.|
|b)||is irreducible iff is an irreducible linear group (generated by linear reflections).|
|c)||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 (hence is a reflection iff one and only one of is a reflection and the others are 1)
b) The "if" part is obvious. Let us prove the "only if" part.
As the group is generated by reflections, the group lies in Therefore, is a completely reducible linear group. Let where are irreducible Consider the subspaces where is an origin, and let be the morphism given by the formula (here is the natural projection). Let Then it is not difficult to check that the map given by the formula defines an equivalence of and
c) Suppose that and are two equivalent groups generated by affine reflections and let establish the equivalence of these groups. Let and be decompositions into products of irreducible groups and let be the corresponding decompositions of the affine spaces and its spaces of translations. We consider and as subgroups of and resp.
It is clear that yields the equivalence of the linear groups and (in the usual sense). Hence is a simple of for every
Let be a reflection. Then for some see a) above. But is also a reflection (inside Therefore for some
The root line of is contained in and this line is where is a root line of Clearly, Hence It follows now from the irreducibility that in fact Therefore (because is generated by all reflections for which has its root line in The same proof is valid for the inverse inclusion. Hence, 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
Let be an irreducible in
One has two possibilities: either is finite or is infinite.
If is finite then there exists a point in which is fixed under Indeed, let be an arbitrary point. Then is fixed under Therefore gives an isomorphism of with see Section (1.1), i.e. is a linear group generated by reflections (we identify and by choosing the origin b).
We shall consider now the case
A beautiful classical theory concerning this case was developed by Coxeter, Witt, Stiefel, see [CRe1961], the results of which we shall recapitulate here.
1) is finite.
is either a Weyl group of an irreducible root system, or a dihedral group, or one of two exceptional groups or
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 is generated by elements which are reflections in the faces of a Weyl chamber There exists a unique set of vectors of unit length with the property: (where in fact and The angle between the mirrors and is of the form The nodes of the Coxeter graph of are in bijective correspondence with the reflections Two nodes and are connected by an edge iff The weight of this edge is equal to (if then the weight is usually omitted). The complete list of finite irreducible is given by the following table of Coxeter graphs. Let us consider the numbers One can change the weight to the numbers 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) is infinite.
is an affine Weyl group of an irreducible root system.
This group is a semidirect product of which is a (finite) Weyl group of a certain root system and the lattice of rank generated by the dual root system The groups thus obtained are distinguished from the others of the list above in the following way (see [Bou1968], Ch. VI):
Theorem. Let be an irreducible finite Then the following properties are equivalent:
|a)||where is an infinite|
|b)||There exists a lattice in of rank|
|c)||is defined over|
|d)||is the Weyl group of a certain irreducible root system, i.e. a group whose Coxeter graph is one of|
|e)||All the numbers lie in|
|f)||The ring with unity, generated over by all of the numbers coincides with|
Let and be an irreducible
1) 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 be a generating system of reflections of We can assume that Therefore this system (and hence is uniquely defined by the system of lines in with the multiplicities
It is well known that an arbitrary set of vectors in 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 (i.e. points of the corresponding projective space).
Proposition. Let and be two sets of lines in and let be arbitrary vectors with For any finite set of indices let us consider the numbers and Then resp. is independent of the choice of the vectors resp. Moreover, the systems and are isometric (i.e. for a certain if and only if for arbitrary
We need only prove that if is fulfilled then for every there exists a number such that for every i.e. the systems and have the same Gram matrix (all the other statements are evident).
Let us fix an index It is not difficult to see that one can assume that the system is "connected", i.e. there exists for every a sequence such that and It is now a matter of straightforward computation to check that one can take
We have seen above that an is defined by a system of lines with the multiplicities It follows from the proposition that such a system is uniquely (up to isometry) defined by a system of numbers (one can derive from these numbers the multiplicities because It will become clear later on why is multiplied by (and not, say by the numbers are of great importance in the whole theory. We call them cyclic products.
So the group (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 i.e. those with all indices distinct, because
We want to point out here several properties of the cyclic products.
a) If is a cyclic permutation of then In other words, depends only on a cycle We use therefore the notation In particular, b) If then 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:
A system of lines with the multiplicities can be described by a graph as follows.
The nodes of the graph are in bijective correspondence with the lines If a node represents the line then this node has the weight Two nodes and are connected by an edge iff and if they are connected then the weight of the edge is equal to It will be convenient to omit the weight of the node, resp. the edge, if it is equal to resp. Moreover, every simple cycle of this graph is supplied with an arbitrary (but fixed) orientation and has weight equal to the corresponding cycle product:
Therefore we now have a way to represent a finite with a fixed generating system of reflections by means of a graph corresponding to the system of lines with the multiciplicities (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 The problem of finding such a method is still unsolved and seems to be very interesting). It is easy to see that, being irreducible, the graph is connected. This graph is called the graph of the group (with respect to a fixed generating system of reflections).
The classification of the finite complex irreducible 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:
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 Any irreducible complex finite (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]):
|The irreducible complex finite|
|No||Type||Graph||Ring generated by cyclic products|
Remark. For those groups among them that are of the form where is an infinite irreducible complex one can obtain an explicit expression for the set of lines from Table 2 below (by taking a vector of unit length in each The explicit expressions of this kind for other groups may either be found in [Coh1976] or be derived directly from the graphs.
2) 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
Let be a point. We identify and by means of see Section 1.1.
Let be a lattice in (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 of This subgroup acts on (see Section 1.1) and it is easy to check that it is an infinite irreducible complex
There are two possibilities: either or If then is not compact, i.e., by definition, is a noncrystallographic group. In this case, can be viewed as a real on
If then is compact, i.e., by definition, is a crystallographic group.
As we know, it is impossible for an infinite real to be noncrystallographic, see Section 1.5.
Denote by a lattice of· equilateral triangles in and by a lattice of squares in Then it is not difficult to check that the groups and are infinite irreducible complex crystallographic
Exercise. Prove that all infinite irreducible complex are equivalent to subgroups of those described in these examples.