ON COMPLEX REFLECTION GROUPS

AND THEIR ASSOCIATED BRAID GROUPS

Michel Broué, Gunter Malle, Raphaël Rouquier

November 1994

Abstract. Presentations “à la Coxeter” are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (still conjectural in some cases) which allow us to generalize some of the known properties of finite Coxeter groups (center of the braid group, construction of Hecke algebras).

1. Background from complex reflection groups

For all the results quoted here, we refer the reader to the classical literature on complex reflections groups, such as [Bou], [Ch], [Co], [ShTo], [Sp], and also to the more recent and fundamental work on the subject by Orlik, Solomon and Terao (see [OrSo], [OrTe]).

Let V be a complex vector space of dimension r. A pseudo–reflection of GL(V) is a non trivial element ρ of GL(V) which acts trivially on a hyperplane, called the reflecting hyperplane of ρ. Let G be a finite subgroup of GL(V) generated by pseudo–reflections. The pair (V, G) is called a “complex reflection group”.

A parabolic subgroup of G is by definition the subgroup of elements of G which act trivially on a subspace of V. By a theorem of Steinberg ([St], Theorem 1.5), a parabolic subgroup is generated by pseudo–reflections.

We denote by A the set of reflecting hyperplanes of G, and we set N := |A|. For H ∈ A, we denote by G_{H} the fixator (pointwise stabilizer) of H (a minimal parabolic subgroup of G), and we set e_{H} := |G_{H}|. We denote by N^{∗} the number of pseudo-reflections in G. The centralizer C_{G}(G_{H}) of G_{H} in G is also its normalizer, as well as the normalizer (setwise stabilizer) of H. We set ω_{H} := |G : C_{G}(G_{H})|.

We denote by S the symmetric algebra of V, by R = S^{G} the algebra of invariants of G, by R_{+} the ideal of R consisting of elements of positive degree, and we set S_{G} := S/R_{+}S.

The following facts are known.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 20, 20G.

This is a summary of work in progress. The final version of this paper will be submitted for publication elsewhere. We thank Jean Michel for useful conversations and for his help in preparing this manuscript.

• There is a family of r integers d_{1}, d_{2}, …, d_{r} called the degrees of (V, G), defined by the following condition: the Poincaré polynomial of the graded module (V ⊗ S_{G})^{G} is

q^{d_{1}−1} + q^{d_{2}−1} + ⋯ + q^{d_{r}−1} .

We have ∑_{i=1}^{i=r}(d_{i} − 1) = ∑_{H∈A}(e_{H} − 1) = ∑_{H∈A/G} ω_{H}(e_{H} − 1) = N^{∗} .

• There is a family of r integers d_{1}^{∗}, d_{2}^{∗}, …, d_{r}^{∗} called the codegrees of (V, G), defined by the following condition: the Poincaré polynomial of the graded module (V^{∗} ⊗ S_{G})^{G} is

q^{d_{1}^{∗}+1} + q^{d_{2}^{∗}+1} + ⋯ + q^{d_{r}^{∗}+1} .

We have ∑_{i=1}^{i=r}(d_{i}^{∗} + 1) = ∑_{H∈A} 1 = ∑_{H∈A/G} ω_{H} = N .

• We have N + N^{∗} = ∑_{i=1}^{i=r}(d_{i} + d_{i}^{∗}) = ∑_{H∈A/G} ω_{H}e_{H} .

• The center Z of G has order |Z| = gcd{d_{1}, d_{2}, …, d_{r}}.

• The order of G is |G| = d_{1}d_{2} ⋯ d_{r}.

Remark. The “codegrees” have not been introduced as such in the quoted literature. Nevertheless, the degrees and the codegrees are related to the exponents {m_{1}, m_{2}, …, m_{r}} and the coexponents {m_{1}^{∗}, m_{2}^{∗}, …, m_{r}^{∗}} (which are defined in [OrSo]) by the formulae

m_{j} = d_{j} − 1 and m_{j}^{∗} = d_{j}^{∗} + 1 (j = 1, 2, …, r) .

2. Presentations

The tables in the Appendix provide a complete list of the irreducible finite pseudo–reflection groups, together with presentations of these groups symbolized by diagrams “à la Coxeter”, as well as some of the data attached to these groups. Many of these presentations were known. This is the case of the rank r groups which are generated by r reflections, studied by Coxeter ([Co]). Some others (the one corresponding to the infinite series) were inspired by [Ar]. We provide here an exhaustive list for the irreduible complex reflection groups.

Here are some definitions, notation, conventions, which will allow the reader to understand the diagrams.

The groups have presentations given by diagrams D such that

• the nodes correspond to pseudo-reflections in G, the order of which is given inside the circle representing the node (order 2 is omitted),

• two distinct nodes which do not commute are related by “homogeneous” relations with the same “support” (of cardinality 2 or 3), which are represented by links beween two or three nodes, or circles between three nodes, weighted with a number representing the degree of the relation (as in the usual case, 3 is omitted, 4 is represented by a double line, 6 is represented by a triple line). These homogeneous relations are called the braid relations of D.

More details are provided in §5 below.

Isomorphisms between diagrams.

We may notice that the only isomorphisms between the diagrams of our tables are between the diagrams of G(2, 1, 2) and G(4, 4, 2), and between the diagrams of 𝔖_{3} and G(3, 3, 2).

Admissible subdiagrams and parabolic subgroups.

Let D be one of the diagrams. Let us define an equivalence relation between nodes by s ∼ s and, for s ≠ t,

s ∼ t : ⟺ s and t are not in a homogeneous relation with support {s, t} .

Then we see that the equivalences classes have 1 or 3 elements, and that there is at most one class with 3 elements.

If there is no class with 3 elements, the rank r of the group is the number of nodes of the diagram, while it is this number minus 1 in case there is a class with

3 elements. Thus s◯_{5} ^{◯^{t}}_{◯ 3 _{u}} has rank 2, as well as ◯_{s}_{4}  ◯_{t}_{3} .

Remark. One must point out that, in the first of the preceding two diagrams, s, t and u must be considered as linked with a line (so t and u do not commute).

An admissible subdiagram is a full subdiagram of the same type, namely a diagram with 1 or 3 elements per class.

Fact. Assume G is neither G_{27}, G_{29}, G_{33} nor G_{34}, and let D be its diagram (see tables).

(1) If D′ is an admissible subdiagram of D, it gives a presentation of the corresponding subgroup G(D′) of G. This subgroup is a parabolic subgroup.

(2) If P_{1} ⊆ P_{2} ⊆ ⋯ P_{n} is a chain of parabolic subgroups of G, there exist g ∈ G and a chain D_{1} ⊆ D_{2} ⊆ ⋯ D_{n} of admissible subdiagrams of D such that

(P_{1}, P_{2}, …, P_{n}) = ^{g}(G(D_{1}), G(D_{2}), …, G(D_{n})) .

Remark.

For groups G_{27} and G_{29}, all isomorphism classes of parabolic subgroups are represented by admissible subdiagrams of our diagrams, but not all conjugacy classes of parabolics subgroups are represented by admissible subdiagrams, as noticed by Orlik.

For groups G_{33} and G_{34}, not all isomorphism classes of parabolic subgroups are represented by admissible subdiagrams of our diagrams. In these cases, it seems that a second diagram should be introduced, as suggested by [Hu].

3. Braid groups and diagrams

We set M := V − ⋃_{H∈A} H , and we denote by p: M → M/G the canonical surjection. Let x_{0} ∈ M. We introduce the following notation for the fundamental groups:

P := π_{1}(M, x_{0}) and B := π_{1}(M/G, p(x_{0})) ,

and we call B and P respectively the braid group and the pure braid group associated to G.

The projection p induces a short exact sequence between fundamental groups

(br)  {1} → P → B → G → {1} .

The following statement is conjectured to be true for all irreducible complex reflection groups, with the diagrams listed in our tables.

It is well known for Coxeter groups (see for example [BrSa] or [De]). Its first assertion had been noticed by Orlik and Solomon ([OrSo]) for the case of Shephard groups (i.e., groups whose braid diagram – see below – is a Coxeter diagram). It is now proved for all the infinite series but G(e, e, r) for e ≥ 2, r > 2, and for all the exceptional groups but G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34}. For the case of non Coxeter-Shephard groups of rank 2, we make use of [Ba].

Theorem–Conjecture.

(1) Let N (D) be the set of nodes of D, identified with a set of pseudo-reflections in G. For each s ∈ N (D), there exists a preimage γ_{s} of s in B such that the set {γ_{s}}_{s∈N (D)}, together with the braid relations of D, is a presentation of B.

(2) The center ZB of B is infinite cyclic and it is generated by an element β which belongs to the monoid generated by the {γ_{s}}_{s∈N (D)}. The length of β on this set of generators is ℓ(β) = (N + N^{∗})/|ZG| .

(3) The short exact sequence (br) induces a short exact sequence between the

centers

{1} → ZP → ZB → ZG → {1} .

Hence the cyclic group ZP is generated by an element π which belongs to the monoid generated by the {γ_{s}}_{s∈N (D)}. Its length on this set of generators is ℓ(π) = N + N^{∗} .

The element β is given in the last column of our tables. Notice that the knowledge of degrees and codegrees allows then to find the order of ZG, which is not explicitely provided in the tables.

The diagram where the orders of the nodes are “forgotten” and where only the braid relations are kept is called a braid diagram for the corresponding group.

Thus the braid diagram  s◯  ^{◯ t}_{◯ u} gives a presentation for the braid

groups of both G(2d, 2, 2) (d ≥ 2), G_{7}, G_{11}, G_{19}, while the diagram s◯_{5} ^{◯^{t}}_{◯_{u}}

gives a presentation for the braid groups of both G_{15} and G(4d, 4, 2).

4. Hecke algebras

Let G be a complex reflection group. We define a set

u = (u_{H,j})_{(H∈A/G)(0≤j≤e_{H}−1)}

of ∑_{H∈A/G}(e_{H}) indeterminates. We set

u^{−1} := (u_{H,j}^{−1})_{(H∈A/G)(0≤j≤e_{H}−1)} .

Now assume that D is a diagram for G, and let s ∈ N (D) be a node of D. We set u_{s,j} := u_{H,j} for j = 0, 1, …, e_{H} − 1, where H denotes the reflecting hyperplane of s.

Definition. The Hecke algebra H_{u} associated to D is the algebra over the ring ℤ[u, u^{−1}] generated by elements (T_{s})_{s∈N (D)} such that

• the elements T_{s} satisfy the braid relations defined by D,

• we have (T_{s} − u_{s,0})(T_{s} − u_{s,1}) ⋯ (T_{s} − u_{s,e_{s}−1}) = 0 .

Notice that through the specialization u_{s,j} ↦ det_{V} (s)^{j} (for s ∈ N (D) and 0 ≤ j ≤ e_{s} − 1), the algebra H_{u} becomes the group algebra of G over a suitable cyclotomic extension ℤ[ζ] of ℤ.

The following statement is well known for Coxeter groups. It is now proved for all infinite series of complex reflection groups, as a consequence of [ArKo], [BrMa], [Ar]. It has been checked for many of the groups of small rank.

Theorem–Conjecture.

H_{u} is a free ℤ[u, u^{−1}]–module of rank |G|.

5. Diagrams and tables

Meaning of the diagrams.

This paragraph provides a list of examples which illustrate the way in which digrams provide presentations for the attached groups.

• The diagram ◯_{s}_{d} ^{e} ◯_{t}_{d} corresponds to the presentation

s^{d} = t^{d} = 1 and \underbrace{ststs ⋯}_{e factors} = \underbrace{tstst ⋯}_{e factors}

• The diagram ◯_{s}_{5}  ◯_{t}_{3} corresponds to the presentation

s^{5} = t^{3} = 1 and stst = tsts .

• The diagram s◯ a e ^{◯ b t}_{◯ c u} corresponds to the presentation

s^{a} = t^{b} = u^{c} = 1 and \underbrace{stustu ⋯}_{e factors} = \underbrace{tustus ⋯}_{e factors} = \underbrace{ustust ⋯}_{e factors} .

• The diagram ◯_{v}⧸◯_{s} ◯_{t} ◯_{w} ◯_{u}⧹ corresponds to the presentation

s^{2} = t^{2} = u^{2} = v^{2} = w^{2} = 1 ,
uv = vu , sw = ws , vw = wv ,
sut = uts = tsu ,
svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

• The diagram s◯ d_{e+1} ^{◯^{t′_{2}}} ⧹_{◯_{t_{2}}} ⧸◯_{t_{3}} corresponds to the presentation

s^{d} = t′_{2}^{2} = t_{2}^{2} = t_{3}^{2} = 1 , st_{3} = t_{3}s ,

st′_{2}t_{2} = t′_{2}t_{2}s ,

t′_{2}t_{3}t′_{2} = t_{3}t′_{2}t_{3} , t_{2}t_{3}t_{2} = t_{3}t_{2}t_{3} , t_{3}t′_{2}t_{2}t_{3}t′_{2}t_{2} = t′_{2}t_{2}t_{3}t′_{2}t_{2}t_{3} ,

\underbrace{t_{2}st′_{2}t_{2}t′_{2}t_{2}t′_{2} ⋯}_{e+1 factors} = \underbrace{st′_{2}t_{2}t′_{2}t_{2}t′_{2}t_{2} ⋯}_{e+1 factors} .

• The diagram e^{^{t′_{2}}◯}⧹_{_{t_{2}}◯}⧸◯_{t_{3}} corresponds to the presentation t′_{2}^{2} = t_{2}^{2} = t_{3}^{2} = 1 , t′_{2}t_{3}t′_{2} = t_{3}t′_{2}t_{3} , t_{2}t_{3}t_{2} = t_{3}t_{2}t_{3} , t_{3}t′_{2}t_{2}t_{3}t′_{2}t_{2} = t′_{2}t_{2}t_{3}t′_{2}t_{2}t_{3} ,

\underbrace{t_{2}t′_{2}t_{2}t′_{2}t_{2}t′_{2} ⋯}_{e factors} = \underbrace{t′_{2}t_{2}t′_{2}t_{2}t′_{2}t_{2} ⋯}_{e factors} .

• The diagram s◯_{5} ^{◯^{t}}_{◯ 3 _{u}} corresponds to the presentation

s^{2} = t^{2} = u^{3} = 1 , stu = tus , ustut = stutu .

• The diagram ◯_{s} △ ◯_{t} ⧸^{◯^{u}} ⧹ ⧹ corresponds to the presentation s^{2} = t^{2} = u^{2} = 1 , stst = tsts , tutu = utut , utusut = sutusu , sus = usu .

• The diagram ◯_{s} △ ◯_{t}^{5} ⧸^{◯^{u}} ⧹ corresponds to the presentation s^{2} = t^{2} = u^{2} = 1 , stst = tsts , tutut = ututu , utusut = sutusu , sus = usu .

• The diagram ◯_{s} ◯_{t} ◯_{u} ⧸^{◯^{v}} ⧹ corresponds to the presentation

\begin{matrix}s^{2} = t^{2} = u^{2} = v^{2} = 1 , sv = vs , su = us , \\ sts = tst , vtv = tvt , uvu = vuv , tutu = utut , vtuvtu = tuvtuv . \end{matrix}

• The diagram s ◯ ^{◯ t}_{◯ u}_{5} 4 corresponds to the presentation s^{2} = t^{2} = u^{2} = 1 , ustus = stust , tust = ustu .

Finally, we denote by a ∧ b the greatest common divisor of two integers a and b.

Information provided by the tables: invariants of braid diagrams.

The groups have been ordered by their diagrams, by collecting groups with the same braid diagram. Thus, for example,

• G_{15} has the same braid diagram as the groups G(4d, 4, 2) for all d ≥ 2,

• G_{4}, G_{8}, G_{16}, G_{25}, G_{32} all have the same braid diagrams as groups 𝔖_{3}, 𝔖_{4} and 𝔖_{5},

• G_{5}, G_{10}, G_{18} have the same braid diagram as the groups G(d, 1, 2) for all d ≥ 2,

• G_{7}, G_{11}, G_{19} have the same braid diagram as the groups G(2d, 2, 2) for all d ≥ 2.

Degrees and Codegrees of a braid diagram.

The following property may be noticed on the tables. It generalizes a property already noticed by Orlik and Solomon for the case of Coxeter–Shephard groups (see [OrSo]).

Theorem. Let D be a braid diagram of rank r. There exist two families of r integers, (d_{1}, d_{2}, …, d_{r}) and (d_{1}^{∗}, d_{2}^{∗}, …, d_{r}^{∗}), depending only on D, and called respectively the degrees and the codegrees of D, with the following property: whenever G is a complex reflection group with D as a braid diagram, its degrees and codegrees are given by the formulae

d_{j} = |Z|d_{j} and d_{j}^{∗} = |Z|d_{j}^{∗} (j = 1, 2, …, r) ,

where |Z| denotes the order of the center of G.

The zeta function of a braid diagram.

In [DeLo], Denef and Loeser compute the zeta function of local monodromy of the discriminant of a complex reflection group G, which is the element of ℚ[q] defined by the formula

Z(q, G) := ∏_{i} det(1 − qμ, H^{i}(F_{0}, ℂ))^{(−1)^{i+1}} ,

where F_{0} denotes the Milnor fiber of the discriminant at 0 and μ denotes the monodromy automorphism (see [DeLo]).

Putting together the tables of [DeLo] and our braid diagrams, one may notice the following fact.

Theorem. The zeta function of local monodromy of the discriminant of a complex reflection group G depends only on the braid diagram of G.

Remark. Two different braid diagrams may be associated to isomorphic braid groups. For example, this is the case for the following rank 2 diagrams (where the sign “∼” means that the corresponding groups are isomorphic) :

For e even, s◯_{e+1} ^{◯^{t}}_{◯_{u}} ∼ s◯ ^{◯ t}_{◯ u} ,

for e odd, s◯_{e+1} ^{◯^{t}}_{◯_{u}} ∼ ◯_{s} ◯_{t},

and s ◯ ^{◯ t}_{◯ u}_{5} 4 ∼ ◯_{s} ◯_{t}.

It should be noticed, however, that the above pairs of diagrams do not have the same degrees and codegrees, nor do they have the same zeta function. Thus, degrees, codegrees and zeta functions are indeed attached to the braid diagrams, not to the braid groups.

name  diagram  degrees  codegrees  β

G(de, e, r)_{e>2,d≥2} [figure: s; ◯ d; e+1; ^{◯^{t′_{2}}}; ⧹; _{◯_{t_{2}}}; ⧸; ◯_{t_{3}}; ◯_{t_{4}}; ⋯; ◯_{t_{r}}] {ed, 2ed … \atop (r−1)ed, rd}  {0, ed, … \atop (r−1)ed} s^{\frac{r}{e∧r}} (t′_{2}t_{2}t_{3} … t_{r})^{\frac{e(r−1)}{e∧r}}

G_{15}  [figure: s; ◯; 5; ^{◯^{t}}; _{◯ 3 _{u}}]  12, 24  0, 24  ustut = s(tu)^{2}

𝔖_{r+1}  ◯_{t_{1}} ◯_{t_{2}}⋯◯_{t_{r}} 2, 3, …, r + 1 0, 1, …,r−1  (t_{1} … t_{r})^{r+1}
G_{4}  ◯_{s}_{3}  ◯_{t}_{3}  4, 6  0, 2  (st)^{3}
G_{8}  ◯_{s}_{4}  ◯_{t}_{4}  8, 12  0, 4  (st)^{3}
G_{16}  ◯_{s}_{5}  ◯_{t}_{5}  20, 30  0, 10  (st)^{3}
G_{25}  ◯_{s}_{3}  ◯_{t}_{3}  ◯_{u}_{3}  6, 9, 12  0, 3, 6  (stu)^{4}
G_{32}  ◯_{s}_{3} ◯_{t}_{3} ◯_{u}_{3} ◯_{v}_{3} 12, 18, 24, 30 0, 6, 12, 18  (stuv)^{5}
G(d, 1, r)_{d≥2} ◯_{s}_{d} ◯_{t_{2}} ◯_{t_{3}}⋯◯_{t_{r}} d, 2d, …, rd 0, d, …, (r−1)d (st_{2}t_{3} … t_{r})^{r}
G_{5}  ◯_{s}_{3}  ◯_{t}_{3}  6, 12  0, 6  (st)^{2}
G_{10}  ◯_{s}_{4}  ◯_{t}_{3}  12, 24  0, 12  (st)^{2}
G_{18}  ◯_{s}_{5}  ◯_{t}_{3}  30, 60  0, 30  (st)^{2}
G_{26}  ◯_{s}  ◯_{t}_{3}  ◯_{u}_{3}  6, 12, 18  0, 6, 12  (stu)^{3}

name  diagram  degrees  codegrees  β

G(2d, 2, r)_{d≥2} s◯ d  ^{◯^{t′_{2}}} ⧹_{◯_{t_{2}}} ⧸◯_{t_{3}} ◯_{t_{4}}⋯◯_{t_{r}}{2d, 4d, … \atop 2(r−1)d, rd} {0, 2d, … \atop 2(r−1)d} s^{\frac{r}{2∧r}} (t′_{2}t_{2}t_{3} … t_{r})^{\frac{2(r−1)}{2∧r}}

G_{7}  s◯  ^{◯ 3 t}_{◯ 3 u}  12, 12  0, 12  stu

G_{11}  s◯  ^{◯ 3 t}_{◯ 4 u}  24, 24  0, 24  stu

G_{19}  s◯  ^{◯ 3 t}_{◯ 5 u}  60, 60  0, 60  stu

G(e, e, r)_{e≥2,r>2} e^{^{t′_{2}}◯}⧹_{_{t_{2}}◯}⧸◯_{t_{3}} ◯_{t_{4}}⋯◯_{t_{r}}  {e, 2e, … \atop (r−1)e, r} {0, e, …, (r−2)e, \atop (r−1)e − r} (t′_{2}t_{2}t_{3} … t_{r})^{\frac{e(r−1)}{e∧r}}
G(e, e, 2)_{e≥3}  ◯_{s} ^{e} ◯_{t}  2, e  0, e − 2  (st)^{e/(e∧2)}
G_{6}  ◯_{s}_{3}  ◯_{t}  4, 12  0, 8  (st)^{3}
G_{9}  ◯_{s}_{4}  ◯_{t}  8, 24  0, 16  (st)^{3}
G_{17}  ◯_{s}_{5}  ◯_{t}  20, 60  0, 40  (st)^{3}
G_{14}  ◯_{s}_{3} ^{8} ◯_{t}  6, 24  0, 18  (st)^{4}
G_{20}  ◯_{s}_{3} ^{5} ◯_{t}_{3}  12, 30  0, 18  (st)^{5}
G_{21}  ◯_{s}_{3} ^{10} ◯_{t}  12, 60  0, 48  (st)^{5}

name diagram  degrees codegrees  β

G_{12}  [figure: s; ◯; ◯t; ◯u]  6, 8  0, 10  (stu)^{4}

G_{13}  [figure: s; ◯; ◯ t; ◯ u; 5; 4]  8, 12  0, 16  (stu)^{3}

G_{22}  s◯ 5 ^{◯t}_{◯u}  12, 20  0, 28  (stu)^{5}
G_{23} ◯_{s} ^{5} ◯_{t}  ◯_{u} 2, 6, 10  0, 4, 8  (stu)^{5}

G_{24}  ◯_{s} △ ◯_{t} ⧸^{◯^{u}} ⧹ ⧹  4, 6, 14  0, 8, 10  (stu)^{7}

G_{27}  ◯_{s} △ ◯_{t}^{5} ⧸^{◯^{u}} ⧹  6, 12, 30  0, 18, 24 (stu)^{5}
G_{28} ◯_{s} ◯_{t} ◯_{u} ◯_{v} 2, 6, 8, 12 0, 4, 6, 10 (stuv)^{6}

G_{29}  ◯_{s} ◯_{t} ◯_{u} ⧸^{◯^{v}} ⧹  4, 8, 12, 20 0, 8, 12, 16 (stuv)^{5}
G_{30} ◯_{s} ^{5} ◯_{t} ◯_{u} ◯_{v} 2, 12, 20, 30 0, 10, 18, 28 (stuv)^{15}

name  diagram  degrees  codegrees  β

G_{31}  ◯_{v}⧸◯_{s} ◯_{t}  ◯_{w} ◯_{u}⧹  {8,12, \atop 20,24}  {0,12, \atop 16,28}  (stuwv)^{6}

G_{33}  ◯_{s} ◯_{t} △ ◯_{u} ⧸^{◯^{w}} ⧹ ◯_{v}  {4,6,10, \atop 12,18}  {0,6,8, \atop 12,14}  (ustvw)^{9}

G_{34}  ◯_{s} ◯_{t} △ ◯_{u} ⧸^{◯^{w}} ⧹ ◯_{v} ◯_{x}  {6,12,18,24, \atop 30,42}  {0,12,18,24, \atop 30,36} (stuvwx)^{7}

G_{35}  ◯_{s_{1}} ◯_{s_{3}} ◯_{s_{4}}^{◯^{s_{2}}} ◯_{s_{5}} ◯_{s_{6}}  {2,5,6,8, \atop 9,12}  {0,3,4,6, \atop 7,10}  (s_{1} … s_{6})^{12}

G_{36}  ◯_{s_{1}} ◯_{s_{3}} ◯_{s_{4}}^{◯^{s_{2}}} ◯_{s_{5}} ◯_{s_{6}} ◯_{s_{7}}  {2,6,8,10,12, \atop 14,18}  {0,4,6,8,10, \atop 12,16} (s_{1} … s_{7})^{9}

G_{37} ◯_{s_{1}} ◯_{s_{3}} ◯_{s_{4}}^{◯^{s_{2}}} ◯_{s_{5}} ◯_{s_{6}} ◯_{s_{7}} ◯_{s_{8}} {2,8,12,14,18,20, \atop 24,30} {0,6,10,12,16,18, \atop 22,28} (s_{1} … s_{8})^{15}

References

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Michel Broué and Raphaël Rouquier : U.F.R. de Mathématiques de l’Université Paris 7 Denis–Diderot et UMR 9994 du CNRS, 2 Place Jussieu, F–75251 Paris Cedex 05, France

   E-mail address: broue@mathp7.jussieu.fr --- rouquier@mathp7.jussieu.fr

Gunter Malle : I.W.R., Im Neuenheimer Feld 368, D–69120 Heidelberg, Germany

   E-mail address: malle@kalliope.iwr.uni-heidelberg.de