Last update: 8 May 2013
We recall some definitions from Looigenga [Loo1980]. Let be a real vector space of finite dimension. A subset of together with an injection is called a root basis if the following conditions are satisfied:
|(i)||(respectively is a linearly independent subset of (resp.|
|(iii)||is a nonpositive integer for|
We order the elements of The Cartan matrix has entries defined by (This deviates from the convention in [Lek1981] and [Loo1980], where
Two root bases and are said to be isomorphic if they are so in the sense of (II.2), i.e. if there exists a linear isomorphism such that and
It follows from proposition (II.2.10) that for each Cartan matrix with properties (C1) and (C2) and integer valued entries there exists a root basis with this Cartan matrix and two such are isomorphic if the vector spaces in which they are defined have the same dimension.
It follows from (II.2.9) that we have and if the root basis is irreducible (see Looiienga [Loo1980]) , the equality holds.
Since condition (C3) of theorem (II.2.10) is automatically satisfied we know that is a Vinberg pair, where
So if as before denotes the subgroup of generated by the reflections and then the conclusions of (II.2.2) to (II.2.6) hold.
But now due to the integrality of the Cartan matrix we have more structure: Let the so-called (root) lattice. Since acts on Let be the group of affine transformations of generated by and This group is called an extended Coxeter group.
(1.2) Remark. In the sequel often will be seen as a subgroup of a non commutative group. Then it is convenient to write an element as For instance In this case we denote the elments by
Since for and we have:
we see that is the semidirect product of and So we have the following two proposition, which are given to compare with the various formulations of the main result of this chapter:
(1.3) Proposition. The group can be described as follows:
Add to the free product of and the following relations:
(1.4) Proposition. The group is generated by where A complete set of relations is
(1.5) Proposition. If then is an affine Coxeter group (5.1). Then it acts simply transitively on the chambers w.r.t. the reflection hyperplanes (mirrors) of
See Bourbaki [Bou1968] Ch. V §3.
The group acts on by the rule for and Let (respectively denote the set of roots (respectively dual roots). This construction is called a generalised root system.
(1.6) Definition. For and we introduce:
(1.7) Proposition. Let and Then
|i)||The quadruple is a hyperplane system (I.2.7),|
Straightforward. For instance vi):
In this chapter the symbols and have a different, although similar meaning as in chapter II. Let have as in §1 a generalised root system and the associated Coxeter (Weyl) group As in (II.3) the group acts properly discontinuously on the domain Also acts properly discontinuously on where the action of is given by for i.e. affects the real plane only.
In the points in regular orbits or the regular points, ??? are by definition the points with trivial stabiliser (of the ???) are the points of the space
This follows from:
(2.2) Lemma. The ??? of non regular points in is the union of the reflection hyperplanes of The reflection hyperplanes are the sets of the form ??? for some and
See Looijenga [Loo1980] lemma 2.17.
Now we define the regular orbit space associated to the extended Coxeter group (or in fact to the generalised root system) by:
In the fundamental chamber (1.1) we choose a point Now (from (2.2)) and we take as a base point for
(2.4) Now we define elements For define:
where is continuous and e.g. see figure 16.
Now and are closed curves (loops) in and they represent by definition the classes and respectively.
Now we can state the main theorem of this chapter:
(2.5) Theorem. The fundamental group of the regular orbit space of an extended Coxeter group is generated by two subgroups isomorphic to and respectively. In turn these groups are generated by and respectively, where and are as above.
Furthermore the following relations hold:
for and where
This is a complete set of relations.
(2.8) Remark. The relations (2.6) are called push relations, because they tell how to "push" through This "pushing" seems to be from the right to the left, however we also have:
|Proof of (2.9).|
By (2.6) the right hand side of (2.9) equals and this equals for
We will prove theorem (2.5) in §3 and §4. Now we first illustrate the elements and the push relations:
Figure 17 depicts a curve in which is the lift of a typical representative of The imaginary part moves from to The real part starts and ends in the origin, but when the imaginary part is on the real part has to be in the layer
Figure 18 depicts the push relations A lift of a typical representative of can be described as follows: First the real part moves from the origin to Then the curve described above takes place, the real part being shifted over So now, if the imaginary part is on the real part is in the layer Therefore intuitively we can see that we can also start The element is then necessary to take care, that the real part ends in the correct point, namely
(2.10) Corollary (cf. prop. (1.3)). The group can be described as follows: Add to the free product of and the push relations (2.6).
As a second reformulation of the main result (2.5) we give an explicit presentation of
(2.11) Corollary (cf. prop. (1.4)). The group is generated by the elements a complete set of relations is:
|1b.||factors both sides.|
In fact these corollaries are equivalent to the theorem.
|Proof of (2.5) (2.11).|
Relations 3b. and 3c. are special cases of the push relations (2.6). Namely, if and then:
Now so If is even, say then
And if is odd, say then
Conversely it is easy to see that the push relations (2.6) follow from these special cases and the commutation relations 2.
Relations 3b. and 3c. say how one can "push through But one cannot push through because for we have so and 3b. becomes This in contrast to where the relation 3a. is valid.
In the special case that for all we get a complete set of relations:
This was conjectured by E. Looijenga in 1978 in a letter to C.T.C. Wall.
It is interesting to see how the presentation of (2.11), similar to what we saw with Coxeter and Artin groups, can be obtained by starting with a presentation of changing it in a suitable equivalent presentation and then deleting some relations. In fact this describes precisely the homomorphism
|A presentation of (1.4):||Another presentation of|
And we see that we get the presentation of corollary (2.11) by deleting the relations:
|3a.||(pushing through is no longer allowed)|
(2.13) Definition. We call the abstract group defined by the presentation (2.11) the extended Artin group associated to the extended Coxeter group The number is called the rank of the extended Artin group. Note that it isn't an extension in the usual sense.
So finally we can rephrase the result as:
(2.14) Theorem. The fundamental group of the regular orbit space associated to an extended Coxeter group is the extended Artin group associated to the extended Coxeter group.
These results were announced in van der Lek [Lek1981].
Let be the canonical surjective homomorphism. In this paragraph we derive a presentation of the subgroup it of as a first step in the proof of the main theorem (2.5) and because it is interesting in itself.
The space equals where and are as defined in (1.6). Recall that below (2.3) we have fixed a point For each we define a so in each chamber a point is given. Now we apply the results of the discussion (II.3.10). So we know that the functor gives us a surjective homomorphism of semigroups
The cover induces an injection of fundamental groups and the image is exactly Hence is isomorphic to Now is canonically isomorphic to as can be seen in the same way as (II.3.16). Let be the surjective homomorphism obtained by and this last isomorphism:
Furthermore we saw in (II.3.12) that is a free semigroup freely generated by the of the one step galleries of the form (II.3.13). In the present situation these one step galleries are of the form:
We denote the of (3.2) by
Now we study the of the cancel and flip relations.
(3.3) Lemma. The of the cancel relations are:
A typical cancel relation is of the form:
Since the of the left element of ($) is
(3.4) Definition. If is a flip relation, then we call flippable and its flipped form. We use the same terminology for
We collect some properties needed in the sequel. Properties (iii), (iv) and (v) are used in §4.
|(i)||A gallery is flippable if and only if there are with (so and the of is of the form:|
|(ii)||In this case the flipped form of the is:|
|(iii)||If and of the form (3.6), then In particular hence and are flippable.|
|(iv)||If the set has the property that then for|
|(v)||If is flippable, then there is a such that|
(i) If is flippable then has the form (3.6). This follows in the same way as (II.3.18) is proved and the fact that is part of the definition of flippable. The converse is obvious.
(ii) From the fact that is the of we know that
(In fact we have etc.) Suppose the flipped form of is factors). We have then that:
(In fact we have etc.) To be able to conclude from (£) and (££) that we have to rule out the possibility that For this consider the galleries of the simple hyperplane system We know that the signed gallery
is flippable and its flipped form is
where (definition (1.6)). It follows that and are the same (positive) dual roots.
(iii) We know that is nonempty so pick an Then for small enough we have for so hence (iii).
(iv) Is obvious.
(v) If is flippable, let be as in (i) and as in the proof of (iii). Note that Set Then is finite, so is an affine group, acting transitively on the chambers w.r.t. its mirrors (see (1.5)). Since is such a chamber and so is So there exists a unique such that Then and because we also have
(3.8) Let and We want to determine explicitly the set of elements such that (3.6) is flippable. Since by (3.5)(ii) we can restrict ourselves to so, since there are four cases: and respectively. The positive dual roots of are linear combinations with integer coefficients and given by the following table:
As is well known. Let us recall one example how these coefficients can be calculated:
Case 3. then
Let be as in (3.5)(i). If then the numbers and satisfy, since and the following:
Conversely, since and are linearly independent the existence of numbers and such that ($) hold, guarantees that Now we look for necessary and sufficient conditions on for the existence of and with ($) in each special case:
(1) and (3) imply so with (2) we see that or so And conversely if the latter is satisfied, there are such that (1), (2) and (3) hold. So
Case 3. and
From (1), (3) and (4) it follows that and from (1), (2) and (3) it follows that So and conversely in each of the four possibilities, there are satisfying (1)-(4). The point can be found in the region indicated in figure 20. Hence
Case 4. and
As in the previous cases it can be derived that where is given by But not all 16 possibilities occur for we have an extra condition, for instance In fact it can be checked that we have the follow: There exist satisfying (1)-(6) if and only if where The 12 open regions in the open square where can be found corresponding to the 12 possibilities of are depicted in Figure 21.
(3.9) Definition. For and we define the elements
It can be checked that where is as defined in (2.4).
(3.10) Corollary of lemma (3.3).
(3.11) Corollary (But in general ??)
We have done the most of the work for the following result:
(3.12) Theorem. The group is generated by A complete set of relations is:
|(2)||where and both sides of (2) have factors and The sets are determined as follows: where|
We have the surjective homomorphism of semigroups (3.1). So the theorem follows from the fact that is freely generated by the elements and the remark that two of galleries have the same if and only if they are equivalent by means of the of the cancel relations and the flip relations. And these are explicitly determined in lemma (3.3) (cancel relations) and the discussion in (3.8) (flip relations).
(3.13) Remark. In the proof of theorem (2.5) given in the next paragraph, we do not use the detailed description of the relations of the group given here. Instead we use properties (iii), (iv) and (v) of proposition (3.5), which we didn't use in this paragraph.
First we outline the proof of theorem (2.5), the main theorem of this chapter. We inbed and in (lemma (4.1) and (4.3)). As long as the proof is not finished, it is important to distinguish between the fundamental group and the abstract group which can be described as follows (cf. def.(2.13) and cor.(2.10)):
Add to the free product of and the push relations (2.6). The inbeddings induce a homomorphism which yields, after we have established the push relations (corollary (4.12)), a homomorphism of the quotient of to The theorem will be proved, once we have shown that this homomorphism is an isomorphism.
To see this we factor the homomorphism via a quotient group of We prove separately that and are isomorphisms (propositions (4.10) and (4.13) respectively). Here are the details:
(4.1) Lemma. The elements as defined in (2.4) and the subgroup of generated by them form an Artin system. In particular is inbedded in
It follows from proposition (3.5)(ii) and (iii) that the elements satisfy the relations and factors on both sides. So there is a homomorphism The image is inside The fact that induces a homomorphism which is a left inverse, so the first homomorphism was injective.
(4.2) Definition. If let be the curve Define
Note that (see (2.4)) hence In view of remark (1.2) the use of the notation needs justification, which is given by the following lemma:
(4.3) Lemma. The map is an injective homomorphism.
It is easy to see that the composition with induces the identity on
Now we prove a prenatal form of the push relations.
(4.4) Proposition. In we have the following relations:
Equivalently: (see figure 18 for this proof)
Let be the curve in defined by
with the property that and Then so is a curve in representing Define curves in by
Then represent and for and 3 respectively. Since
the curve is defined and represents the left hand side of (4.6). So it suffices to prove that
For this we check with the help of formulas (4.7) that the following hold:
Property (i) follows from and for we have Moreover for the latter is contained in but then for all because we have so also:
so This proves (i). (ii) is proved analogously and (iii) follows from (4.8).
(4.9) Definition. Let be the group which can be described as follows: Add to the free product of and the relations (4.5).
The inclusion and the inbedding (4.3) induce a homomorphism and by (4.4) this gives a homomorphism
(4.10) Proposition. is an isomorphism.
Since we have so is surjective. By the relations (4.5), which by definition (4.9) hold in it follows that Now let and and consider Suppose Then is the unique decomposition of in so Hence Now is already bijective on as can be seen form the diagram. So
Before we state the next proposition, we recall the definition of the elements for and
Since and these elements can be interpreted as elements of one of the groups and finally The elements can be seen only as elements of the last three groups (unless or So the statement has only in this three cases a meaning. The statement is not true in
(4.11) Proposition. In and we have
(4.12) Corollary. In and the push relations (2.6) hold.
By definition the push relations (2.6) hold in The fact that they hold in implies, that the inclusion induces a homomorphism
(4.13) Proposition. This homomorphism is an isomorphism.
As pointed out earlier this proposition together with proposition (4.10) proves theorem (2.5). For the proof of (4.13) we need some preparations.
(4.14) Definition. We define an action, called pseudo conjugation, of on as follows: for and
Note that is not a homomorphism. Instead we have
In the same way we can define an action of on as subgroup of where is the projection induced by the projection and the identity After the establishing of the theorem and can be identified. This action of corresponds to an action of (shifting) on which is defined in the following lemma:
(4.16) Lemma. An action of on is induced by For and we have:
Let We have to prove that for If for then for some and Then Property (4.17) follows from
Since is freely generated by we have a homomorphism of semigroups such that Denote the composition with by so
(4.18) Lemma. the functor is equivariant w.r.t the action and restricted to i.e. for and we have:
First we prove: Suppose the lemma is true for and and let Then the lemma is true for
So we are done if we check the lemma for a one step gallery:
On the other hand:
and by (2.9) this equals also
(4.19) Proposition. There exists a unique homomorphism such that we have Note that
It is enough to check that the elements satisfy the cancel relations and the flip relations. First the cancel relations:
Now the flip relations: Suppose and and Let and (both factors). We have to prove: Let be as exists by (3.5)(v), i.e. Then it follows of lemma (4.18) and proposition (3.15)(iv) that ? Since because the flip relations hold by definition in we have hence also But is injective on as follows from the diagram, so Hence
|Proof of proposition (4.13).|
The homomorphism induces a homomorphism and this induces a homomorphism which is an inverse of as easily can be checked on various generators.
Now theorem (2.5) is completely proved.
In this paragraph we make first some remarks about affine or parabolic linear Coxeter groups. This makes clear why the result of Nguyen Viet Dung [Ngu1983] is indeed a special case of theorem (II.3.8). Next we establish an isomorphism between an extended Artin group of finite type and a certain affine Artin group.
(5.1) Definition. A Coxeter group is called affine or parabolic, if it is isomorphic to a group of affine transformations of an affine space such that acts properly discontinuously on and is generated by orthogonal reflections.
It can be proved (Bourbaki [Bou1968]) that such a group is generated by the reflections in the walls of a chamber with respect to the reflection hyperplanes of the reflections in the group. The group forms with these generators a Coxeter system.
If one want to construct a space of regular orbits in the situation of definition (5.1) there are various possibilities (depending on a parameter which we will encounter in (5.3). One possibility (corresponding to is obvious by the affine nature of Let and let the group act on by acting on both components. The complexified reflection hyperplanes are then of the form where is a real affine reflection hyperplane of This is the approach of Nguyen Viet Dung [Ngu1983].
It turns out that it is not important to distinguish between these various complexifications, because they lead to homeomorphic regular orbit spaces, as we will see in (5.3).
(5.2) Definition. Let be a linear Coxeter group acting on a real vector space is called an affine (or parabolic) linear Coxeter group is there is an invariant hyperplane with an invariant innerproduct and
Let such that and Then is also invariant and acts on as an affine Coxeter group (in particular as described in def. (5.1). Conversely given an affine Coxeter group as in (5.1) we can form considering as a vector space. Now each (affine) reflection in extends uniquely to a linear reflection in and in this way we get a parabolic linear Coxeter group. The form can be defined by
(5.3) Complexification. Let be an affine linear Coxeter group in a space and let be as before. Let and be the space of regular points and the regular orbit space respectively of as defined in (II.3.2) and (II.3.3).
Let and consider Let and The map is a homeomorphism equivariant under the action of so is homeomorphic to On the other hand for we have that is a deformation retract of because if then and This deformation retraction is also equivariant for the action of so is a deformation retract of
(5.4) In particular this shows that the result of Nguyen Viet Dung, which in fact says that the fundamental group of is the Artin group associated to is a special case of theorem (II.3.8).
We apply these ideas to the situation of an extended Coxeter group of finite type, i.e. is finite. Then the number of roots is finite and we have a classical root system. We assume this to be irreducible. Let be the longest dual root, the corresponding root. The corresponding reflection is defined by for Define also the affine reflection by Let be the group of affine transformations of generated by Then it is known (Bourbakl [Bou1968] Ch.VI §2) that is a (parabolic) Coxeter system, is a fundamental domain of and We want to establish an isomorphism between the (parabolic) Artin group associated to and the extended Artin group
(5.5) Theorem. Let be the root basis of a finite irreducible root system. Let be the longest dual root, the corresponding root, the corresponding reflection in the Weyl group
Let be the Artin system associated to the Coxeter system where
Then and are isomorphic and an isomorphism is induced by:
where and is the image of under the injection discussed in (II.3.?).
From the fact that is finite it follows that is an affine Coxeter group acting on as discussed in the beginning of this paragraph. We construct the (affine) linear Coxeter system as suggested there. Put
And as usual: for Now leaves invariant for each Let us in particular study this action for
For we have:
So the action of on corresponds with the action of on Hence So by the reasoning in (5.3) we have hence
We check that an isomorphism is explicitly given by (5.6). Consider the diagram
where ($) is the deformation retraction defined in (5.3) and ($$) is the multiplication with The basepoint of is its image under But the basepoint of is So we define and then is a curve from the basepoint of to the of the basepoint of So an explicit isomorphism is given by
Now let be a curve in such that represents the element so for instance we can take with Let Then which is a curve from to Now we have to study the curve where For note that so represents the right hand side of (5.7). The curve is illustrated in Figure 22, where the root system is shown.
Now we make four remarks about this curve These remarks are sufficient to describe its homotopy type:
From these considerations it follows that is a loop in representing the element So the image of is A similar reasoning gives that the image of is
(5.8) Corollary. If is an extended Artin group of rank such that is finite and irreducible, then has a finite presentation with generators.
The conclusion of this corollary is true under some other conditions too:
(5.9) Theorem. If is an extended Artin group of rank with for all and with a connected Dynkin diagram, then has a finite presentation with generators.
If then (2.9), so then can be eliminated. Since the Dynkin diagram is connected all but one of the generators can be eliminated.
But there is an important difference between (5.8) and (5.9). The relations of the presentation meant in (5.8) are pretty simple, because they are the relations of an Artin group. But the relations become very complicated if one follows the procedure of the proof of (5.9).
This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.