Transformation Groups, Vol. 1, No. 3, 1996, pp. 1–19  ◯^{c} Birkhäuser Boston (1996)

PRESENTATIONS FOR CRYSTALLOGRAPHIC COMPLEX REFLECTION GROUPS

GUNTER MALLE^{∗}

Universität Heidelberg I.W.R., Im Neuenheimer Feld 368, D–69120 Heidelberg, Germany

malle@urania.iwr.uniheidelberg.de

Abstract. We find presentations for the irreducible crystallographic complex reflection groups W whose linear part is not the complexification of a real reflection group. The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2]. As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part. We then classify the reflections in the groups W and the minimal number of them needed to generate W, using the diagrams. Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits of W can be derived from our presentations.

1. Introduction

There has been some interest in finding “nice” presentations for the finite irreducible (complex) reflection groups, starting with the work of Coxeter for real reflection groups, then continued by Coxeter and Shephard for certain finite complex reflection groups. A complete list of presentations for the irreducible finite complex reflection groups was given in [2]. Here we give presentations for irreducible infinite discrete complex reflection groups. These groups were classified by Popov [8]. In the real case, one finds the affine Weyl groups. Presentations for these can be obtained from the socalled extended Dynkin diagrams.

It turns out that an analogous construction works in the complex case: Hughes [6] defined extended root-diagrams for certain classes of irreducible finite complex reflection groups. For these groups, our presentations can be

Typeset by AMS-TEX

∗The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft.

Received February 7, 1996, revised version received July 11, 1996. Accepted July 16, 1996.

derived from the latter diagrams by adding a single further relation. In the general case, the infinite irreducible complex reflection group W is presented with a Coxeter-type diagram that contains at most two more nodes than the diagram in [2] presenting the finite linear part Lin(W) of W.

Given the presentations, it is easy to derive some further properties of the infinite irreducible complex reflection group in question, like the number of classes of reflections and the structure of W/W′.

In the final section we show that at least in some cases our presentation for an irreducible infinite discrete complex reflection group W allows to obtain a presentation for the fundamental group of the space of regular orbits of W. A corresponding result is well known in the finite real case by work of Brieskorn and Deligne, in most finite complex cases by [3], and in the real affine case by Viêt Dũng. It would be interesting to prove this in general, and without case-by-case arguments. This would in particular require an a-priori way to define the nice presentation of a complex reflection group.

2. Crystallographic complex reflection groups

The finite complex reflection groups have been classified by Shephard and Todd. Nice presentations for these were obtained in [1] and [2], completing earlier work of Coxeter, Shephard and others. Here we are interested in the infinite affine reflection groups. The following setup is taken from the lecture notes [8] of Popov.

Let E be the n-dimensional affine space over ℂ, and V its space of translations. By choosing a base point v_{0} ∈ E we may identify the group A(E) of affine transformations of E with the semi-direct product V ·GL(V). For W a subgroup of A(E), the image of W in the factor group GL(V) is denoted by Lin(W); it is independent of the choice of v_{0}. The subgroup of translations of W is denoted by Tran(W).

We fix a positive definite sesquilinear (i.e., unitary) form ⟨ , ⟩ on V. An element of finite order σ ∈ A(E) is called an affine reflection if it preserves the distance on V induced by ⟨ , ⟩ and the codimension of its space of fixed points is equal to 1. A subgroup W of A(E) is called an r-group if it is discrete and generated by affine reflections.

By [8, Theorem 1.4] every r-group is a direct product of irreducible rgroups, so it suffices to consider the irreducible groups. It is clear that any finite r-group fixes some point of E and thus is equivalent to a finite complex reflection group in the usual sense. The infinite irreducible r-groups W were classified by Popov. There arise two essentially different cases. Either E/W is not compact in which case W is equivalent to the complexification of an affine Weyl group (loc. cit., Theorem 2.2). If E/W is compact, W is called crystallographic. Here quite a number of new cases arise (i.e., cases without real analogue).

The extension of Tran(W) by Lin(W) defines an action of Lin(W) by automorphisms of the lattice Tran(W), hence a real representation of Lin(W) on the tensor product Tran(W) ⊗_{ℤ} ℝ. Thus, for crystallographic W, if Lin(W) is the complexification of a real reflection group, then this representation will be reducible, but irreducible otherwise.

Definition 2.1. A genuine crystallographic complex reflection group is an irreducible crystallographic complex reflection group W for which Lin(W) is not the complexification of a real reflection group.

In the sequel, we will be concerned with the genuine crystallographic complex reflection groups.

Proposition 2.2. Let W be a genuine crystallographic complex reflection group. Then the center Z(W) is trivial.

Proof. A nontrivial central element in Tran(W) gives rise to an invariant subspace of dimension 1 in the corresponding real representation of Lin(W). Since for genuine groups this representation is irreducible of dimension at least 2, Z(W) ∩ Tran(W) = 1. But no element with nontrivial image in the natural quotient Lin(W) can be central in W since the reflection representation of Lin(W) on V is faithful and V is the linear span of Tran(W). □

3. The presentations

The presentations for the crystallographic complex reflection groups will be given by means of suitably generalized Coxeter diagrams D, in the spirit of [2]. A presentation for a group W is encoded in a diagram consisting of small circles (nodes), of edges between two of the nodes, of big circles and half circles connecting precisely three nodes and of inscribed ∗’s and triangles. More precisely, we use the following conventions: The small circles in a diagram stand for the generators of W. The order of a generator is given by the number inscribed in the circle, except for the case of order 2, where the number is omitted.

The nodes in a diagram are always numbered from left to right, and from bottom to top if two of them are in the same column.

If two nodes are neither directly joined by a line nor by a big circle, the corresponding generators commute. If they are joined by a single line, they satisfy the ordinary braid relation στσ = τστ. A double bond stands for στστ = τστσ, a triple bond for (στ)^{3} = (τσ)^{3}. An edge marked with the sign ∞ means: do not add any relation between σ and τ. The big circle and the double big circle

[figure: ◯; ◯; ◯; ◯; ◯; ◯]

stand for the circular relations σ_{1}σ_{2}σ_{3} = σ_{2}σ_{3}σ_{1} = σ_{3}σ_{1}σ_{2}, respectively σ_{1}σ_{2}σ_{3}σ_{1} = σ_{2}σ_{3}σ_{1}σ_{2} = σ_{3}σ_{1}σ_{2}σ_{3}, between the three nodes involved, while the half circle

[figure: ◯; ◯; ◯]

stands for the two relations σ_{1}σ_{2}σ_{3} = σ_{2}σ_{3}σ_{1} and σ_{1}σ_{2}σ_{3}σ_{2} = σ_{3}σ_{2}σ_{3}σ_{1}. The small triangle inside the big triangle

◯ △ ◯ ⧸◯⧹  or ◯ △ ◯ ⧸ ⧸◯⧹

signifies the additional relation (σ_{1}σ_{2}σ_{3})^{2} = (σ_{3}σ_{1}σ_{2})^{2} between the three generators. Finally, the ∗ inside the triangle

◯ ◯⧹ ⧹ ⧸ ⧸ ◯ ∗

stands for the further relation (σ_{1}σ_{2}σ_{1}σ_{3})^{2} = (σ_{3}σ_{1}σ_{2}σ_{1})^{2}, and in

◯⧸◯ ⧹◯ ∗ ⧹ ⧸◯

for the additional relation σ_{2}σ_{3}σ_{4}σ_{1}σ_{2}σ_{3}σ_{4}σ_{1}σ_{2} = σ_{3}σ_{4}σ_{1}σ_{2}σ_{3}σ_{4}σ_{1}σ_{2}σ_{3}.

It turns out that in all cases in order to obtain a diagram for W one can take the diagram for Lin(W) in loc. cit., adjoin one or two nodes corresponding to highest roots (as introduced for example in [6]), and then add a single further relation. The results are given in the form of tables.

Theorem 3.1. The Tables I and II give presentations for the genuine crystallographic complex reflection groups W. Here the generators all correspond to reflections of W.

Here, in the case of variable dimension n, the diagram has n + 1 nodes, unless it involves a big circle, when there are n+2 generators. In Tables I and II, the linear part Lin(W) is obtained by omitting the [ ] and possible lower and upper indices. In Table I, the group Lin(W) is imprimitive; the dimension is hence given by the third parameter n in the brackets (m, p, n).

Proof. The proof consists of calculations with generators and relations. Let W be a genuine crystallographic reflection group. Then its structure as an extension of Tran(W) by Lin(W) is described in [8, Table 2]. In most cases this extension is split. A nice presentation for the finite irreducible reflection group Lin(W) is given in [2]. We thus obtain a first presentation for W by adjoining to the presentation for Lin(W) a basis of the lattice Tran(W) as additional generators, and the relations describing the action of Lin(W) on Tran(W), plus the relations which force Tran(W) to be commutative. Clearly, the additional generators are no reflections, and the total number of generators is now at least 3 dim(Lin(W)). In the next step, the additional generators need to be replaced by suitable reflections while carefully keeping track of the relations.

We start with the infinite families. First, let W = [G(4, 1, n)]_{1} in the notation of [8]. Then W is the extension of the finite irreducible complex reflection group L = Lin(W) ≅ G(4, 1, n) by a suitable lattice T = Tran(W). The n-dimensional group G(4, 1, n) has a presentation (see [1])

◯ 4 ◯ ⋯ ◯

with respect to the n generators given by the matrix σ_{1} := diag(i, 1, …, 1) (here i := \sqrt{−1}) together with the permutation matrices σ_{j} for the transpositions (j − 1, j), 2 ≤ j ≤ n. The lattice T is generated by the vectors

ρ_{2j−1} := e_{j}, ρ_{2j} := −ie_{j}, for 1 ≤ j ≤ n,

where {e_{1}, …, e_{n}} denotes the standard basis of V = ℂ^{n} (see [8, Table 2]). From this, we may easily deduce the action of L on T. If we set

σ̃_{n+1} := σ_{n} ⋯ σ_{2}ρ_{1}σ_{1}σ_{2} ⋯ σ_{n},

then all the ρ_{j} may be expressed in terms of σ_{1},…, σ̃_{n+1}; more precisely, we have

\begin{matrix}ρ_{2j−1} = & σ_{j+1} ⋯ σ̃_{n+1} ⋯ σ_{2}σ_{1}^{−1}σ_{2} ⋯ σ_{j}, \\ ρ_{2j} = & σ_{j} ⋯ σ_{2}σ_{1}^{−1}σ_{2} ⋯ σ̃_{n+1} ⋯ σ_{j+1} \end{matrix}

for 1 ≤ j ≤ n. By direct computation it follows that σ_{1}, …,σ̃_{n+1} satisfy the relations given by the diagram on n + 1 nodes

◯ 4 ◯ ⋯ ◯ ◯ 4  with (σ_{1} ⋯ σ̃_{n+1} ⋯ σ_{2})^{2}.  (3.1)

On the other hand, it is also easy to verify that all the relations between the σ_{1}, …, σ_{n}, ρ_{1}, …, ρ_{2n} are consequences of the relations implied by the above diagram. Thus the diagram (3.1) gives one possible presentation for W. Replacing σ̃_{n+1} by σ_{n+1} := σ̃_{n+1}σ_{n} ⋯ σ_{1} ⋯ σ_{n} this presentation is transformed to the one occurring in the Table I. Note that σ_{n+1} = σ_{n} ⋯ σ_{2}ρ_{1}σ_{1}^{2}σ_{2} ⋯ σ_{n}, and since ρ_{1}σ_{1}^{2} is a reflection, all generators in the final presentation are in fact reflections. This shows the assertion for type [G(4, 1, n)]_{1}. To continue we need the following lemma:

Lemma 3.2. The various genuine crystallographic reflection groups with Lin(W) ≅ G(4, p, n), p ∈ {1, 2, 4}, can be realized as reflection subgroups of [G(4, 1, n)]_{1} as follows:

\xymatrix{[G(4, 4, n)] \ar@{^{(}->}[d] \ar@{^{(}->}[r] & [G(4, 2, n)]_{1} \ar@{^{(}->}[d] \ar@{^{(}->}[r] & [G(4, 1, n)]_{2} \ar@{^{(}->}[d] \\ [G(4, 2, n)]_{1}^{∗} \ar@{^{(}->}[r] & [G(4, 2, n)]_{2} \ar@{^{(}->}[r] & [G(4, 1, n)]_{1}} .  (3.2)

For each arrow, the index of the corresponding groups is equal to 2. The group [G(4, 4, n)] is generated by

{σ_{2}, σ_{1}σ_{2}σ_{1}^{−1}, σ_{3}, …, σ_{n}, σ_{n+1}σ_{n}σ_{n+1}}

in the notation introduced above. The groups to the right are obtained from [G(4, 4, n)] by adding σ_{1}^{2}, σ_{1} respectively, while the ones in the lower row are obtained from those in the upper row by adjoining σ_{n+1}.

Proof. This is easily checked from the description of the groups in [8, Table 2]. For example, from loc. cit. it is clear that [G(4, 1, n)]_{2} is the subgroup of [G(4, 1, n)]_{1} of index 2 generated by σ_{1}, …, σ_{n} together with ρ_{1}ρ_{3}^{−1}, say. A short calculation shows that this subgroup is also generated by σ_{1}, …, σ_{n}, σ_{n+1}σ_{n}σ_{n+1}. Similarly, [G(4, 2, n)]_{2} is the subgroup of index 2 of [G(4, 1, n)]_{1} generated by

{σ_{1}^{2}, σ_{2}, σ_{1}σ_{2}σ_{1}^{−1}, σ_{3}, …, σ_{n+1}}.

The only infinite series of groups for which the extension of Tran(W) with Lin(W) is nonsplit are the ones denoted by [G(4, 2, n)]_{1}^{∗}. It can be read off from [8, Table 2] that [G(4, 2, n)]_{1}^{∗} is the subgroup of [G(4, 2, n)]_{2} generated by the elements

{ρ_{1}ρ_{3}^{−1}, ρ_{1}σ_{1}^{2}, σ_{1}σ_{2}σ_{1}^{−1}, σ_{2}, …, σ_{n+1}}.

Since

\begin{matrix}ρ_{1}ρ_{3}^{−1} = σ_{2} ⋯ σ_{n+1} ⋯ σ_{2}σ_{3} ⋯ σ_{n+1} ⋯ σ_{3}σ_{1}^{2}σ_{2}σ_{1}^{2}σ_{2}, \\ ρ_{1}σ_{1}^{2} = σ_{2} ⋯ σ_{n+1} ⋯ σ_{2}, \end{matrix}

the first two generators are superfluous. □

To complete the proof of the theorem for the infinite series of groups with Lin(W) = G(4, p, n), p ∈ {1, 2, 4}, we can just apply the Reidemeister-Schreier method (see for example [7, Th. 2.8]) to the subgroups of finite index of the finitely presented group [G(4, 1, n)]_{1} listed in Lemma 3.2 to obtain the desired result.

For groups of type [G(6, m, n)], m ∈ {1, 2, 3, 6}, we start with W = [G(6, 1, n)]. The n-dimensional finite complex reflection group Lin(W) ≅ G(6, 1, n) has a presentation (see [1])

◯ 6 ◯ ⋯ ◯

with respect to the n generators given by the matrix σ̃_{1} := diag(−ζ, 1, …, 1) (here ζ := exp(2πi/3)) together with the permutation matrices σ̃_{j} for the transpositions (j −1, j), 2 ≤ j ≤ n. The lattice T is generated by the vectors {ρ_{2j−1} := e_{j}, ρ_{2j} := ζe_{j} | 1 ≤ j ≤ n}. If we set

σ̃_{n+1} := σ̃_{n} ⋯ σ̃_{2}ρ_{1}σ̃_{1}^{2}σ̃_{2} ⋯ σ̃_{n},

then a short calculation shows that W is generated by σ̃_{1}, …, σ̃_{n+1}, and a presentation on these generators is given by

◯ 6 ◯ ⋯ ◯ ◯ 3  with (σ̃_{1} ⋯ σ̃_{n+1} ⋯ σ̃_{2})^{2}.  (3.3)

We now replace the original generators σ̃_{j} by σ_{j} := σ̃_{n+2−j} for 1 ≤ j ≤ n, and σ_{n+1} := σ̃_{1} ⋯ σ̃_{n+1} ⋯ σ̃_{2} to obtain the presentation given in the table. The argument for the case Lin(W) ≅ G(6, p, n), p ∈ {2, 3, 6}, can now be completed by the Reidemeister-Schreier method, using the following result which is again easily deduced from [8, Table 2].

Lemma 3.3. The various genuine crystallographic reflection groups with Lin(W) ≅ G(6, p, n), p ∈ {2, 3, 6}, can be realized as reflection subgroups of [G(6, 1, n)] as follows:

\xymatrix{[G(6, 6, n)] \ar@{^{(}->}[d] \ar@{^{(}->}[r] & [G(6, 2, n)]_{1} \ar@{^{(}->}[d] \\ [G(6, 3, n)]_{1} \ar@{^{(}->}[r] & [G(6, 1, n)]}  .  (3.4)

Here the index in the rows is equal to 3, in the columns equal to 2. The group [G(6, 6, n)] is generated by

{σ_{2}, σ_{1}σ_{2}σ_{1}^{−1}, σ_{3}, …, σ_{n}, σ_{n+1}σ_{n}σ_{n+1}}

in the notation introduced above. The groups to the right are obtained by adding σ_{1}, the ones in the lower row from those in the upper row by adjoining σ_{n+1}.

Literally the same arguments work for the groups [G(3, m, n)]_{1,2}, m ∈ {1, 3}, hence we omit the details.

For the exceptional cases, the approach taken for the imprimitive cases works at least for those W with Lin(W) of dimension at most 3. For the larger cases one could try to prove the assertion with the help of a computer. Nevertheless, in order to write down an algorithm which checks that all relations obtained by an explicit construction of W as extension of Tran(W) by Lin(W) are consequences of the relations given in Table II one would almost have to solve the word problem in these finitely presented groups. It is not clear how that should be done. Therefore, some more subtle approach is needed.

First let W = [K_{31}], so Lin(W) ≅ K_{31}. According to [1] this group has a presentation

◯ ◯ ◯ ⧸◯  ◯⧹

on generators

σ_{1} :=[\begin{matrix}i & 1 + i & 0 & 1 \\ 1 − i & −i & 0 & −1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] , σ_{2} :=[\begin{matrix}−1 & −1 + i & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] ,

σ_{3} :=[\begin{matrix}1 & 0 & 0 & 0 \\ −1 − i & −1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] , σ_{4} :=[\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] ,

σ_{5} :=[\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & −1 \end{matrix}] ,

while Tran(W) is generated by {ρ_{2j−1} := e_{j}, ρ_{2j} := ie_{j} | 1 ≤ j ≤ 4}, where i := \sqrt{−1}. Then W is generated by σ_{1}, …, σ_{5} together with

σ_{6} := σ_{5}σ_{1}σ_{3}σ_{1}σ_{4}σ_{3}σ_{1}ρ_{7}σ_{5}σ_{1}σ_{3}σ_{4}σ_{1}σ_{3}σ_{1}σ_{5}.

Further, these elements satisfy all the relations of the group G with presentation given by the diagram in the row for [K_{31}] in Table II. In order to show that the kernel of the corresponding epimorphism ϕ : G → W is trivial we proceed as follows. Using the computer algebra system GAP [9] it can be verified in a few seconds that the subgroup H_{1} of G generated by {σ_{1}, σ_{2}, σ_{4}, σ_{5}, σ_{6}} has index 12. Further, a presentation for H_{1} on these generators can be derived in GAP with the Reidemeister-Schreier method and using Tietze transformations for simplification. It turns out that H_{1} is the finitely presented group described by the row for [K_{29}] in Table II. Next, the subgroup H_{2} := ⟨σ_{1}, σ_{2}, σ_{5}, σ_{6}, σ_{4}σ_{2}σ_{1}σ_{2}σ_{4}⟩ of H_{1} has index 10 in H_{1}, and its presentation on these generators is the one of [G(4, 4, 4)] proved to be correct previously. Thus, the kernel of ϕ|_{H_{2}} is trivial. Further, by explicit calculation with the matrices given above it is verified that ϕ(H_{2}) contains a subgroup of index 2 of Tran(W). Since Lin(ϕ(H_{2})) ≅ G(4, 4, 4) has index 5 in Lin(ϕ(H_{1})) ≅ K_{29} this proves that ϕ|_{H_{1}} is an isomorphism. Finally, Lin(ϕ(H_{1})) ≅ K_{29} has index 12 in Lin(ϕ(G)) ≅ K_{31}, so that ϕ itself has to be an isomorphism.

The matrices for σ_{2}, …, σ_{6} generate the reflection group [K_{31}]^{∗}. Using GAP one finds that the subgroup generated by these elements has index 16 in G, and has the presentation given in Table II. The groups [K_{32}], [K_{33}] and [K_{34}] can be dealt with on a computer using the implementation of the Knuth-Bendix algorithm by D. Holt [5]. □

4. Extended root diagrams

The result can be reformulated in terms of “extended root diagrams”. For this let W be a finite irreducible complex reflection group and S a standard set of generators as in [4] or [2]. The elements of S will be called simple reflections. For each simple reflection modulo conjugation choose an eigenvector for the nontrivial eigenvalue. The set R of images of these eigenvectors under the action of W is then called a set of roots for W (in analogy to the case of real reflection groups, see also [4, 6]). This is partitioned into the disjoint union of roots corresponding to reflections chosen from some fixed conjugacy class. Let C be such a class and R_{C} the subset of roots for reflections belonging to C. Any v ∈ R_{C} can be written as the image under some w ∈ W of some v_{0} belonging to a simple reflection. The height of v is then the minimum of the lengths of elements w as above, written as products of simple reflections. A root of maximal height in R_{C} is called a highest root.

Hughes [6] studied the root diagrams obtained from the root diagram of W by adjoining some highest root in the case that W is generated by n reflections, where n denotes the dimension of W. By comparing with his results we can reformulate part of Theorem 3.1:

Corollary 4.1. Let W̄ be an n-dimensional finite irreducible complex reflection group generated by n reflections. Then presentations for the genuine crystallographic reflection groups W with Lin(W) ≅ W̄ are given by the diagrams obtained by adjoining some highest root to the diagram for W̄, together with a single further relation.

Note that in the case of finite real Weyl groups W̄ a presentation of the corresponding affine Weyl group W is also obtained by adjoining a highest root to the diagram for W̄ (but without having to add any further relations).

5. Number of generating reflections

For a complex reflection group W let m(W) denote the minimal number of reflections needed to generate W. If W is finite irreducible in dimension n then n ≤ m(W) ≤ n+1, and the precise value is known in all cases [10]. We can now determine m(W) for genuine crystallographic complex reflection groups.

Proposition 5.1. Let W be a genuine crystallographic complex reflection group. Then m(W) = m(Lin(W)) + 1, unless

• the extension of Tran(W) by Lin(W) is nonsplit or W = [G(6, p, n)]_{1}, p = 2, 3, when m(W) = m(Lin(W)), or

• W = [G(4, 2, 2)]_{1}, when m(W) = m(Lin(W)) + 2 = 5. In particular the number of reflections needed to generate W is at least n+1 and at most n + 2, except for [G(4, 2, 2)]_{1} which is generated by 5 = n + 3 reflections, where n denotes the dimension of Lin(W).

Proof. Let W be a genuine n-dimensional crystallographic reflection group and assume that n reflections σ_{1}, …, σ_{n} generate W. Then their images in the quotient Lin(W) are generating reflections of Lin(W) in its reflection representation on V. Since the representation of Lin(W) is irreducible, the intersection of the hyperplanes in V fixed by the images of the σ_{i} consists only of the origin. Therefore, the intersection of the hyperplanes H_{i} in E pointwise fixed by the σ_{i} consists of exactly one point. Hence the group generated by σ_{1}, …, σ_{n} fixes a point of E and thus is finite. So at least n + 1 reflections are needed to generate W. An upper bound on the number of generators is given by the presentations in Theorem 3.1. This proves the proposition unless m(Lin(W)) = n+1, W ∉ {[G(6, 2, n)]_{1}, [G(6, 3, n)]_{1}} and the extension of Tran(W) by Lin(W) is split.

So either W = [G(4, 2, n)]_{1,2,(3)} or W ∈ {[G(6, 2, 2)]_{2}, [G(6, 3, 2)]_{2}, [K_{12}], [K_{31}]}. In the cases W = [G(4, 2, 2)]_{j} the commutator factor group of W is elementary abelian of order 2^{k}, with k = 4 for j = 2, 3, k = 5 for j = 1, so any generating set has to contain at least k elements, which gives the necessary lower bound on m(W). Now let W = [G(4, 2, n)]_{1,2} with n ≥ 3. Let σ_{1}, …, σ_{n+1} be n + 1 reflections in W and σ̄_{1}, …, σ̄_{n+1} their images in W̄ := Lin(W) ≅ G(4, 2, n). The group W̄ has a quotient isomorphic to the symmetric group S_{n}, with m(S_{n}) = n − 1. Thus we may assume that σ̄_{1}, …, σ̄_{n−1} map to transpositions in S_{n}. Further, if W̄ is generated by σ̄_{1}, …, σ̄_{n+1} then not both σ̄_{n} and σ̄_{n+1} can lie in the kernel of the epimorphism onto S_{n}. Since W̄ has just two classes of reflections this means that σ̄_{n}, say, also has to be conjugate to σ̄_{j}, j ≤ n − 1. Since ⟨σ_{1}, …, σ_{n}⟩ is finite by the above argument, we also have ⟨σ_{1}, …, σ_{n}⟩ ≅ ⟨σ̄_{1}, …, σ̄_{n}⟩, which shows that σ_{1}, …, σ_{n} are already conjugate in W. In the case W = [G(4, 2, n)]_{2} this proves that W cannot be generated by σ_{1}, …, σ_{n+1} since W/W′ is elementary abelian of order 2^{3}. If W = [G(4, 2, n)]_{1} we at least conclude that ⟨σ̄_{1}, …, σ̄_{n}⟩ is isomorphic to either G(4, 4, n) or to G(2, 2, n). Moreover we may assume without loss of generality that σ̄_{j}, j ≤ n, are the standard generators of G(4, 4, n), G(2, 2, n) respectively. Since G(4, 2, n) has just two classes of generators it follows that we may moreover assume that σ_{1}, …, σ_{n+1} satisfy the relations of a suitable subset of the n + 2 generators in the diagram for [G(4, 2, n)]_{1}. But it can easily be checked that any proper subset of the standard generators for [G(4, 2, n)]_{1} generates a proper subgroup not isomorphic to W.

The remaining four exceptional cases can be dealt with by similar arguments which shall be omitted. □

6. Conjugacy classes of reflections

In this section we determine the conjugacy classes of reflections in genuine crystallographic reflection groups. More precisely, we classify reflections up to the relation that the corresponding hyperplanes of fixed points lie in the same orbit under W.

Proposition 6.1. Table III gives the number of orbits of reflecting hyperplanes, the orders of their pointwise stabilizers, and the structure (as a direct product of cyclic groups) of the commutator factor group W/W′ for the genuine crystallographic complex reflection groups.

Proof. Let W be a genuine crystallographic reflection group and σ ∈ W an affine reflection. Then the image σ̄ of σ in Lin(W) under the canonical epimorphism is again a reflection in Lin(W). Representatives of the classes of reflections in the finite groups Lin(W) are known. When determining the conjugacy classes of affine reflections σ ∈ W we may thus assume that σ̄ is one of those representatives.

Let now W = [G(4, 1, n)]_{1}. Via the identification of A(E) with V ·GL(V) we embed W into GL_{n+1}(ℂ). The generators σ_{1}, …, σ_{n} of W are then mapped to diag(1, i, 1, …, 1) and to the permutation matrices for the transpositions (j, j + 1), 2 ≤ j ≤ n, while ∏_{j=1}^{2n} ρ_{j}^{a_{j}} is represented by the matrix

[ \begin{matrix}1 & 0 \\ a & I_{n} \end{matrix} ] ,

where a = (a_{1} − ia_{2}, …, a_{2n−1} − ia_{2n})^{t} and I_{n} is the n × n-identity matrix.

Let σ ∈ W be an affine reflection. Then σ̄ is conjugate to either σ_{1}^{k}, 1 ≤ k ≤ 3, or to σ_{2}. Assume first that σ̄ = σ_{1}. Then

σ = [ \begin{matrix}1 & 0 & 0 \\ b_{1} & i & 0 \\ b & 0 & I_{n−1} \end{matrix} ] ,

with b_{1} ∈ ℤ + iℤ and b = (0, …, 0)^{t} since σ has finite order. Conjugating by suitable elements from Tran(W) one finds that without loss of generality b_{1} ∈ {0, 1}. Thus, there are at most two classes of reflections σ ∈ W with σ̄ = σ_{1}. On the other hand an easy calculation shows that in fact the two representatives are not conjugate inside W. Similarly, if σ̄ = σ_{1}^{2}, then σ is conjugate to an element of the form

[ \begin{matrix}1 & 0 & 0 \\ b_{1} & −1 & 0 \\ 0 & 0 & I_{n−1} \end{matrix} ] ,

where b_{1} ∈ {0, 1, 1 + i}. If b_{1} ∈ {0, 1 + i}, then σ lies inside the cyclic group generated by a suitable reflection of order 4, thus we get one new conjugacy class of hyperplanes for b_{1} = 1. Finally, if σ̄ = σ_{2}, then up to conjugation the preimage σ is unique. This proves the assertion for W = [G(4, 1, n)]_{1}.

For the remaining W = [G(4, m, n)] a descent argument can be used. By Lemma 3.2 any such W can be reached from W_{0} := [G(4, 1, n)]_{1} through a chain of normal subgroups with relative index 2. Any reflection in W lies inside some class of reflections of W_{0}, thus it suffices to study the decomposition of the classes of W_{0} along the chain of normal subgroups. We omit the details.

Similar arguments work for the other cases. □

By inspection of the list of reflecting hyperplanes obtained in the preceding proof the following corollary can be deduced. It will be needed in the sequel.

Corollary 6.2. Let W_{1} ≤ W_{2} be genuine crystallographic complex reflection groups such that W_{1} is a reflection subgroup of W_{2} of finite index. Then W_{1} and W_{2} have the same sets of reflecting hyperplanes if and only if

(W_{1}, W_{2}) ∈ {([G(4, 2, n)]_{2}, [G(4, 1, n)]_{1}), ([G(4, 2, n)]_{1}, [G(4, 1, n)]_{2})} .

Remark. If W is a finite complex reflection group, then it is known that the invariants of the abelian group W/W′ and the sizes of the centralizers of reflecting hyperplanes, counted once for every orbit of such hyperplanes, coincide (see for example [4, (1.8)]). The result of Proposition 6.1 shows in particular that this does not hold in general for infinite complex reflection groups. Nevertheless it still remains true in a considerable number of cases. The reason for the failure of this result in general is the fact that in an affine complex reflection group the product of two non-commuting reflections may again be a reflection; this can not happen in a finite reflection group.

7. Braid groups

Let W be a genuine crystallographic reflection group on the affine space E. Then by definition the space E/W of orbits is compact, and it is connected with respect to the quotient topology. Let R = R(W) denote the set of (affine) reflections in W, and for r ∈ R, let H_{r} be the (affine) subspace of fixed points of r. Define

E^{·} := E \ ⋃_{r∈R} H_{r},  Ē := E^{·}/W,

the complement E^{·} in E of the reflecting affine hyperplanes and its quotient Ē by W. The topological space Ē is connected, and for a chosen base point x_{0} ∈ Ē we write π_{1}(Ē, x_{0}) for its fundamental group, or, by abuse of notation, π_{1}(W).

In the case of a finite real reflection group, it was shown by Brieskorn and Deligne that a presentation for the fundamental group π_{1} can be obtained from the Coxeter presentation of W by omitting the order relations. A similar statement is true for finite complex reflection groups, starting with the presentations given in [2], up to finitely many possible exceptions (see [3]). If W is an infinite real reflection group, then it is known to be a Coxeter group, and the corresponding statement for the fundamental group was shown by Viêt Dũng [11]. The case of infinite complex reflection groups seems to be open.

In Theorem 3.1 we gave a presentation on reflections for each irreducible crystallographic complex reflection group W for which Lin(W) is not the complexification of a real group. This presentation consists of braid relations, described by edges, non-edges or circles in a diagram, or by the additional ∗ or triangle, and of order relations (the one additional relation should also be thought of being an order relation). We define the braid group B(W) associated to W to be the finitely presented group with generators in 1-1 correspondence with the generators of W, but subject only to the braid relations.

By comparing the underlying topological spaces to the spaces of regular orbits for affine Weyl groups and then invoking the result of Viêt Dũng we obtain the following partial result on the structure of π_{1}(W), covering seven of the thirteen infinite series:

Theorem 7.1. Let W be a genuine crystallographic complex reflection group of type

\begin{matrix}[G(3, 1, n)], [G(4, 1, n)]_{1}, [G(4, 1, n)]_{2}, [G(4, 2, n)]_{1}, \\ [G(4, 2, n)]_{2}, [G(6, 1, n)], or [G(6, 2, n)]_{1} . \end{matrix}

Then its fundamental group is isomorphic to its braid group, i.e.,

B(W) ≅ π_{1}(W).

Proof. Let us first consider the 1-dimensional case W = [G(e, 1, 1)] with e ∈ {3, 4, 6}. Here the groups W are the so-called groups of genus 1 and Ē is homeomorphic to a sphere with three points deleted. For example, if e = 4 then W is the group of translations T with respect to the lattice Λ = ℤ+iℤ, extended by the group of rotations Lin(W) of order 4 generated by the multiplication by i. The quotient (ℂ \ Λ)/T is homeomorphic to a torus minus four points, and Ē is homeomorphic to a sphere minus three points, i.e., to ℂ \ {0, 1}. In particular, the fundamental group is a free group on two generators, and thus isomorphic to B(W).

We next consider W = [G(e, 1, n)]_{(1)}. We let E_{n} denote the underlying affine space. Let C̃_{n} be the affine Weyl group of type C_{n}. The idea is to show that the space Ē for W is homeomorphic to the correspondingly formed space for C̃_{n}. For this let W^{∗} be the kernel of the natural epimorphism κ : W → S_{n} isomorphic to Lin(W) extended by the homocyclic abelian group of order e^{n}. Thus (E_{n}, W^{∗}) is the direct product of n copies of (E_{1}, [G(e, 1, 1)]), and the space E^{∗} obtained by removing from E_{n} the reflecting hyperplanes for reflections r ∈ W^{∗} equals the direct product of n copies of the space E_{1}^{·} for n = 1. In particular, E^{∗}/W^{∗} is homeomorphic to the direct product of n copies of ℂ \ {0, 1}. The quotient W/W^{∗} ≅ S_{n} acts on this by permutation of the coordinates, and Ē is homeomorphic to

{(z_{1}, …, z_{n}) ∈ ℂ^{n} | z_{j} ≠ 0, 1, z_{j} ≠ z_{k} for j ≠ k}/S_{n}.

But this is the same space as the underlying space for C̃_{n}, hence B(W) is isomorphic to the fundamental group in this real affine case whose structure is known by [11].

By Corollary 6.2 we know that W = [G(4, 1, n)]_{1} and its reflection subgroup W_{0} = [G(4, 2, n)]_{2} of index 2 have the same set of reflecting hyperplanes. In particular the spaces E^{·} agree, the space Ē(W_{0}) is a two-fold covering of Ē(W) and π_{1}(W_{0}) is a subgroup of index 2 of π_{1}(W). Write σ_{1}, …, σ_{n+1} for the generators of π(W), ordered along the diagram, and let κ̃ : π_{1}(W) → S_{n} be the composite of the canonical epimorphism from π_{1}(W) to W with the map κ : W → S_{n} introduced before. By the previous identification of π_{1}(W) with π_{1}(C̃_{n}) it follows that κ̃ maps σ_{2}, …, σ_{n} to the standard generators of S_{n}, while σ_{1}, σ_{n+1} have trivial image. Since the restriction of κ̃ to π_{1}(W_{0}) is still surjective, and all σ_{i}, 2 ≤ i ≤ n, are conjugate in π_{1}(C̃_{n}), it follows that π_{1}(W_{0}) also contains σ_{2}, …, σ_{n}. It is now an easy exercise to show that all subgroups of index 2 of π_{1}(C̃_{n}) with this property have a presentation of type B(W_{0}). This proves the statement for type [G(4, 2, n)]_{2}.

Now let W = [G(4, 1, n)]_{2} with translation subgroup

Tran(W) = {∑_{j=1}^{n}(a_{2j−1} + ia_{2j})e_{j} | a_{j} ∈ ℤ, ∑_{j=1}^{2n} a_{j} ≡ 0 (mod 2)}

and complement Lin(W)=G(4, 1, n). Then W′:=(∑_{j=1}^{n}(ℤ+iℤ)e_{j})·Lin(W) is the reflection group [G(4, 1, n)]_{1} containing W as a normal subgroup of index 2. Let E^{·} = E^{·}(W), Ē := E^{·}/W and Ē′ := E^{·}/W′. Thus Ē is a two-fold covering of

Ē′ = {(z_{1}, …, z_{n}) ∈ ℂ^{n} | z_{j} ≠ 0, z_{j} ≠ z_{k} for j ≠ k}/S_{n}  (7.1)

(see above) unramified outside the image of the hyperplanes defined by z_{i} = 1. The same observations apply to the space of regular orbits of the affine Weyl group of type B̃_{n}, considered as a reflection subgroup of index 2 inside C̃_{n}. Hence by the result of [11] for π_{1}(B̃_{n}) we get that π_{1}(W) ≅ π_{1}(B̃_{n}) ≅ B(W). Now by Corollary 6.2 the group [G(4, 1, n)]_{2} and its reflection subgroup [G(4, 2, n)]_{1} of index 2 have the same sets of reflecting hyperplanes. Hence a descent argument as for [G(4, 2, n)]_{2} proves the assertion about the fundamental group π_{1}([G(4, 2, n)]_{1}).

Finally we consider W = [G(6, 2, n)]_{1}, a reflection subgroup of index 2 of W′ := [G(6, 1, n)]. With E^{·} = E^{·}(W), Ē := E^{·}/W and Ē′ := E^{·}/W′ as above we see that again Ē is a two-fold cover of Ē′ which is homeomorphic to the one given in (7.1). The result again follows. □

The result in Theorem 7.1 raises the following question:

Question 7.2. Given a genuine crystallographic complex reflection group W, is it true that the fundamental group π_{1}(W) is isomorphic to the braid group B(W) as defined above from the presentation of W given in Theorem 3.1 ?

Table I.A. Imprimitive Lin(W)

W (see [8])  D  additional relation
[G(3, 1, 1)]  ◯ 3 ^{∞} ◯ 3  (σ_{1}σ_{2})^{3}
[G(3, 1, n)]_{1}, n ≥ 2 ◯ 3 ◯ ⋯ ◯ ◯ 3  (σ_{1} ⋯ σ_{n+1} ⋯ σ_{2})^{3}

[G(3, 1, 2)]_{2}  ◯ 3 △ ◯ ⧸ ⧸◯⧹  (σ_{1}σ_{2}σ_{3})^{3}

[G(3, 1, n)]_{2}, n ≥ 3 ◯ 3 ◯ ⋯ ◯ △ ◯ ⧸◯⧹  (σ_{1} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{2})^{3}
[G(3, 3, 3)]  tetrahedron^{1}  (σ_{1}σ_{2}σ_{1}σ_{3}σ_{4}σ_{3})^{3}

[G(3, 3, n)], n ≥ 4 ◯ △ ◯ ⧸◯⧹ ⋯ ◯ △ ◯ ⧸◯⧹  (σ_{1} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{3})^{3}
[G(6, 1, 1)]  ◯ 3 ^{∞} ◯  (σ_{1}σ_{2})^{6}
[G(6, 1, n)], n ≥ 2 ◯ 3 ◯ ⋯ ◯ ◯  (σ_{1} ⋯ σ_{n+1} ⋯ σ_{2})^{6}
[G(6, 2, 2)]_{1}  ◯ ◯ 3 ◯  (σ_{1}σ_{2}σ_{3})^{6}
[G(6, 2, 2)]_{2}  ◯  ◯ ◯ 3 ⧹ ⧸ ⧸◯  (σ_{2}σ_{3}σ_{4})^{3}

[G(6, 2, n)]_{1}, n ≥ 3 ◯ 3 ◯ ⋯ ◯ ◯ ⧸◯  (σ_{1} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{2})^{6}

[G(6, 3, 2)]_{1}  ◯ ∗ ◯ ⧸ ⧸◯⧹  (σ_{1}σ_{2}σ_{3})^{6}

[G(6, 3, 2)]_{2}  [figure: ◯; ◯; ◯; ⧹; ⧸; ⧸; ◯]  (σ_{1}σ_{2}σ_{3}σ_{4})^{4}

[G(6, 3, n)]_{1}, n ≥ 3 ◯ △ ◯ ⧸◯⧹ ⋯ ◯ ◯  (σ_{1} ⋯ σ_{n+1} ⋯ σ_{3})^{6}
[G(6, 6, 3)]  ◯⧸ ⧹△◯ ◯ ⧹ ⧸◯ △  (σ_{1}σ_{2}σ_{3}σ_{4})^{6}

[G(6, 6, n)], n ≥ 4  ◯ △ ◯ ⧸◯⧹ ⋯ ◯ ◯ ⧸◯  (σ_{1} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{3})^{6}

1Here ”tetrahedron” stands for the presentation on four generators σ_{1}, …, σ_{4} subject to relations σ_{i}σ_{j}σ_{i} = σ_{j}σ_{i}σ_{j} for all i < j, (σ_{i}σ_{j}σ_{k})^{2} = (σ_{k}σ_{i}σ_{j})^{2} for all i<j < k.

16 PRESENTATIONS FOR COMPLEX REFLECTION GROUPS

Table I.B

W (see [8])  D  additional relation
[G(4, 1, 1)]  ◯ 4 ^{∞}◯  (σ_{1}σ_{2})^{4}
[G(4, 1, n)]_{1}, n ≥ 2  ◯ 4 ◯ ⋯ ◯ ◯  (σ_{1} ⋯ σ_{n+1} ⋯ σ_{2})^{4}
[G(4, 1, 2)]_{2}  ◯ ◯ 4 ◯  (σ_{1}σ_{2}σ_{3})^{4}
[G(4, 1, n)]_{2}, n ≥ 3  ◯ 4 ◯ ⋯ ◯ ◯ ⧸◯  (σ_{1} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{2})^{4}
[G(4, 2, 2)]_{1}  [figure: ◯; ◯; ◯; ◯; ◯; ∞; ∞]  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5})^{2}
[G(4, 2, 3)]_{1}  ◯  ◯ ◯ ◯ ⧹ ◯ ⧸  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5})^{4}
[G(4, 2, n)]_{1}, n ≥ 4 ◯  ◯ ◯⧹ ⧸◯ ⋯ ◯ ◯ ⧸◯  (σ_{1}σ_{2}σ′_{2} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{3})^{4}
[G(4, 2, 2)]_{1}^{∗}  ◯ ∗ ◯ ⧸ ⧸◯⧹ ⧹  (σ_{1}σ_{2}σ_{3})^{4}
[G(4, 2, n)]_{1}^{∗}, n ≥ 3  ◯ △ ◯ ⧸ ⧸◯⧹ ⋯ ◯ ◯  (σ_{1} ⋯ σ_{n+1} ⋯ σ_{3})^{4}
[G(4, 2, 2)]_{2,3}  ◯  ◯ ◯⧹ ⧹ ⧸ ⧸◯  (σ_{1}σ_{2}σ′_{2}σ_{3})^{4}
[G(4, 2, n)]_{2}, n ≥ 3 ◯  ◯ ◯⧹ ⧸◯ ⋯ ◯ ◯ (σ_{1}σ_{2}σ′_{2} ⋯ σ_{n+1} ⋯ σ_{3})^{4}
[G(4, 4, 3)]  ◯⧸ ⧹ △◯ ◯⧹ ⧸◯ △  (σ_{1}σ_{2}σ_{3}σ_{4})^{4}
[G(4, 4, n)], n ≥ 4  ◯ △ ◯ ⧸ ⧸◯⧹ ⋯ ◯ ◯ ⧸◯  (σ_{1} ⋯ σ_{n+1}σ_{n−1} ⋯ σ_{3})^{4}

G. MALLE 17

Table II. Primitive Lin(W)

W (see [8]) dim  D  add. relation
[K_{4}]  2  ◯ 3 ◯ 3 ⧸◯ 3 ⧹  (σ_{1}σ_{2}σ_{3})^{2}
[K_{5}]  2  ◯ 3 ◯ 3 ◯ 3  (σ_{1}σ_{2}σ_{3})^{3}
[K_{8}]  2  ◯ 4 ◯ 4 ◯ 4  (σ_{1}σ_{2}σ_{3})^{4}
[K_{12}]  2  [figure: ◯; ◯; ◯; ⧹; ⧸; ◯]  (σ_{1}σ_{2}σ_{3}σ_{4})^{3}
[K_{12}]^{∗}  ◯ ◯ ⧹ ⧸◯ ∗  (σ_{1}σ_{2}σ_{3})^{8}
[K_{24}]  3  ◯ ∗ ◯ ⧸ ⧸◯⧹  ◯  (σ_{1}σ_{2}σ_{3}σ_{4})^{7}
[K_{25}]  3  ◯ 3 ◯ 3 ⧸◯ 3 ⧹◯ 3  (σ_{1}σ_{2}σ_{3}σ_{4})^{3}
[K_{26}]_{1}  3  ◯ 3 ◯ 3 ◯ 3 ◯  (σ_{1}σ_{2}σ_{3}σ_{4})^{4}
[K_{26}]_{2}  ◯ 3 ◯ 3 ◯ ◯  (σ_{1}σ_{2}σ_{3}σ_{4})^{6}
[K_{29}]  4  ◯ ◯ △ ◯ ⧸ ⧸◯⧹ ◯  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5})^{8}
[K_{31}]  4  ◯ ◯ ◯ ⧸◯  ◯⧹ ◯  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5}σ_{6})^{5}
[K_{31}]^{∗}  ◯ ◯ ◯ △ ◯ ⧸ ⧸◯⧹  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5})^{12}
[K_{32}]  4 ◯ 3 ◯ 3 ◯ 3 ◯ 3 ◯ 3  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5})^{6}
[K_{33}]  5  ◯ ◯⧸◯ ⧹◯ ∗ ⧹ ⧸◯ ◯  (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5}σ_{6})^{6}
[K_{34}]  6 ◯⧸◯ ⧹◯ ∗ ⧹ ⧸◯ ◯ ◯ ◯ (σ_{1}σ_{2}σ_{3}σ_{4}σ_{5}σ_{6}σ_{7})^{12}

18 PRESENTATIONS FOR COMPLEX REFLECTION GROUPS

Table III. Orbits of reflecting hyperplanes, and W/W′

W (see [8])  #orbits  orders  W/W′
[G(3, 1, 1)]  3  3, 3, 3  3 × 3
[G(3, 1, n)]_{1}, n ≥ 2  4  3, 3, 3, 2  3 × 3 × 2
[G(3, 1, n)]_{2}, n ≥ 2  2  3, 2  3 × 2
[G(3, 3, n)], n ≥ 3  1  2  2
[G(4, 1, 1)]  3  4, 4, 2  4 × 2
[G(4, 1, n)]_{1}, n ≥ 2  4  4, 4, 2, 2  4 × 2 × 2
[G(4, 1, 2)]_{2}  4  4, 2, 2, 2  4 × 2 × 2
[G(4, 1, n)]_{2}, n ≥ 3  3  4, 2, 2  4 × 2
[G(4, 2, 2)]_{1}  6  2, 2, 2, 2, 2, 2  2 × 2 × 2 × 2 × 2
[G(4, 2, n)]_{1}, n ≥ 3  3  2, 2, 2  2 × 2
[G(4, 2, 2)]_{1}^{∗}  4  2, 2, 2, 2  2 × 2 × 2
[G(4, 2, n)]_{1}^{∗}, n ≥ 3  3  2, 2, 2  2 × 2
[G(4, 2, 2)]_{2,3}  5  2, 2, 2, 2, 2  2 × 2 × 2 × 2
[G(4, 2, n)]_{2}, n ≥ 3  4  2, 2, 2, 2  2 × 2 × 2
[G(4, 4, n)], n ≥ 3  1  2  2
[G(6, 1, 1)]  3  6, 3, 2  3 × 2
[G(6, 1, n)], n ≥ 2  4  6, 3, 2, 2  3 × 2 × 2
[G(6, 2, 2)]_{1}  4  3, 3, 2, 2  3 × 2 × 2
[G(6, 2, 2)]_{2}  3  3, 2, 2  3 × 2 × 2
[G(6, 2, n)], n ≥ 3  3  3, 3, 2  3 × 2
[G(6, 3, n)]_{1}, n ≥ 2  3  2, 2, 2  2 × 2
[G(6, 3, 2)]_{2}  2  2, 2  2 × 2
[G(6, 6, n)], n ≥ 3  1  2  2
[K_{4}]  1  3  3
[K_{5}]  2  3, 3  3 × 3
[K_{8}]  2  4, 2  4
[K_{12}]  1  2  2
[K_{12}]^{∗}  1  2  2
[K_{24}]  1  2  2
[K_{25}]  1  3  3
[K_{26}]_{1,2}  2  3, 2  3 × 2
[K_{29}]  1  2  2
[K_{31}]  1  2  2
[K_{31}]^{∗}  1  2  2
[K_{32}]  1  3  3
[K_{33}]  1  2  2
[K_{34}]  1  2  2

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