ON THE REGULARITY OF THE FINITE REFLECTION GROUPS

R. B. Howlett ^{∗} and Jian-yi Shi † ∗† School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

† Department of Mathematics, East China Normal University, Shanghai, 200062, P. R. China

Abstract. We show that an irreducible finite reflection group G of rank greater than 1 is regular if and only if the associated root system is perfect. This holds if and only if G is a Coxeter group. Hence we get a characterization of Coxeter groups among all the irreducible finite reflection groups of rank greater than 1.

The irreducible finite unitary reflection groups were classified by Shephard and Todd ([16]) and by Cohen ([4]). They have been studied extensively by many people since then (see [2; 3; 10; 11; 12; 13; 14; 15; 17]). Finite Coxeter groups are a special family of reflection groups, whose properties are relatively well known. It is interesting to ask what properties of Coxeter groups are shared also by the other reflection groups, and what are not. In the present paper, we shall consider a property, which we call regularity, of a reflection group, defined by the existence of a basic section of the associated root system R (see 3.1). We shall show that an irreducible finite reflection group G of rank > 1 is regular if and only if all the root line circles in R are perfect (see 2.3 for the definitions). Then we further show that this holds if

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The second named author is grateful for the financial support of the Australian Research Council and for the hospitality of the University of Sydney.

2 R. B. Howlett and J. Y. Shi

and only if G is a Coxeter group. Thus we get a characterization of Coxeter groups among all the finite reflection groups.

Let G be a finite reflection group which acts irreducibly on a hermitian space V, and let S be a minimal set of reflections which generates G. Our strategy for proving our main result is as follows. We show in Lemma 1.7 that the order |Z(G)| of the center Z(G) of G divides the cardinality of any root line in R. We further show in Lemma 4.2 (2) that if G is regular then |Z(G)| divides the orders o(s) of all s ∈ S. By the classification of the irreducible finite complex reflection groups, there are only a few irreducible finite complex reflection groups G satisfying this property (see Proposition 4.3). On the other hand, in each of these exceptional irreducible complex reflection groups G, we can find at least one root line circle which is imperfect (see 4.5-4.11). This proves our main result.

The contents of the paper are organized as follows. In Section 1 we collect some definitions and results concerning irreducible finite reflection groups G which are either well known or easily proved. Then in Section 2 we introduce the concept of perfectness for a root line circle, a root system and a reflection group. Lemma 2.6, concerned with perfectness, is crucial in the proof of our main result. Regularity is introduced in Section 3, where we establish the equivalence of regularity of G and perfectness of its associated root system R (see Theorem 3.4). Our main result, Theorem 4.12, is proved in Section 4.

§1. Roots and reflections. We collect some definitions and results concerning irreducible finite reflection groups; many of them follow from Cohen’s paper [4]. 1.1. Let V be a complex vector space of dimension n. A reflection in V is a linear transformation of V of finite order with exactly n − 1 eigenvalues equal to 1. A reflection group G in V is a finite group generated by reflections

Finite Reflection Groups 3

in V. The group G is reducible if it is a direct product of two proper reflection subgroups and irreducible otherwise. The action of G on V is said irreducible if V has no nonzero proper G-invariant subspace. In the present paper we shall always assume that G is irreducible and acts irreducibly on V. A reflection group G in V is called a real group or a Coxeter group if there is a G-invariant ℝ-subspace V_{0} of V such that the canonical map ℂ ⊗_{ℝ} V_{0} → V is bijective. If this is not the case, G will be called complex. (Note that, according to this definition, a real reflection group is not complex.)

There exists a unitary inner product ( , ) on V invariant under G; hence we may assume that G is a subgroup of U(V), the group of all unitary transformations with respect to the unitary inner product. From now on we will make this assumption. 1.2. A root of a reflection in V is an eigenvector corresponding to the unique nontrivial eigenvalue of the reflection. A root of G is a root of a reflection in G.

Let s be a reflection in V of order d > 1. There is a vector a ∈ V of length 1 and a primitive d-th root ζ of unity such that s = s_{a,ζ}, where s_{a,ζ} is defined by

(1.2.1)  s_{a,ζ}(v) = v + (ζ − 1)(v, a)a

for all v ∈ V. We also write s_{a,d} for s_{a,ζ} if ζ = e^{2πi/d}.

We use the notation |x| for the cardinality of x if x is a set, and for the absolute value of x if x is a complex number. The meaning will always be clear from the context.

For each v ∈ V define o_{G}(v) to be the order of the (necessarily cyclic) group consisting of the identity and the reflections in G which have v as a root. (This group is G_{W} = { g ∈ G | gu = u for all u ∈ W }, where W = v^{⊥}.)

4 R. B. Howlett and J. Y. Shi

Thus o_{G}(v) > 1 if and only if v is a root of G. If a is a root of G, then the number o_{G}(a) will be called the order of a (with respect to G). We shall denote o_{G}(a) simply by o(a) when G is clear from the context.

Note that we shall also use the notation o(ζ) for the order of ζ, where ζ could be either a root of unity, or a group element. This should cause no confusion. Lemma 1.3. o(gv) = o(v) = o(cv) for all v ∈ V, g ∈ G and c ∈ ℂ^{∗}, where ℂ^{∗} := ℂ \ {0}. Proof. Let W = v^{⊥}. Then we have

u ∈ W ⟺ (u, v) = 0 ⟺ (gu, gv) = 0 ⟺ gu ∈ (gv)^{⊥}.

This implies gW = (gv)^{⊥}. Now we have

h ∈ G_{gW} ⟺ h(gu) = gu for all u ∈ W,
⟺ (g^{−1}hg)u = u for all u ∈ W,
⟺ g^{−1}hg ∈ G_{W} ,
⟺ h ∈ ^{g}G_{W} .

So we get G_{gW} = ^{g}G_{W} and hence o(gv) = |G_{gW} | = |^{g}G_{W} | = |G_{W} | = o(v). The equation o(v) = o(cv) follows from the fact that v^{⊥} = (cv)^{⊥}. □ 1.4. A pair (R, f) is called a root system in V, if

(i) R is a finite set of vectors of V of length 1; (ii) f : R → ℕ\{1} is a map such that s_{a,f(a)}R = R and f(s_{a,f(a)}(b)) = f(b)

for all a, b ∈ R; (iii) the group G generated by { s_{a,f(a)} | a ∈ R } is a (finite) reflection group,

and for all a ∈ R and c ∈ ℂ,

ca ∈ R ⟺ ca ∈ Ga.

Finite Reflection Groups 5

The group G is called the reflection group associated with the root system (R, f). We have o_{G}(a) = f(a) for any a ∈ R.

We shall denote a root system (R, f) simply by R when f is clear from the context. 1.5. A system of simple roots is a pair (B, w), where B is a finite set of vectors of V and w is a map from B to ℕ \ {1}, satisfying the following conditions:

(i) for all a, b ∈ B, we have |(a, b)| = 1 ⟺ a = b; (ii) the group G generated by S = { s_{a,w(a)} | a ∈ B } is finite; (iii) there is a root system (R, f) with R = GB and f(a) = w(a) for all

a ∈ B; (iv) for any B′ ⫋ B, the group G′ generated by { s_{a,w(a)} | a ∈ B′ } is a

proper subgroup of G. We call the elements of S simple reflections, and the subgroups of G

generated by the subsets of S standard parabolic subgroups of G (relative to S). We also call (R, f) the root system of G generated by B, and B a simple system for R.

Note that we do not require B to be linearly independent. If B is linearly independent, then condition (iv) holds automatically.

The above definition of simple system is considerably weaker the usual definition for Coxeter groups; in particular, it is not true that whenever B_{1} and B_{2} are simple systems for the same root system R then there is an element g ∈ G with gB_{1} = B_{2}.

It is easily seen that G acts irreducibly on V if and only if the root system R (resp. the simple root system B) spans V and cannot be decomposed into a disjoint union of two proper subsets R_{1} and R_{2} (resp. B_{1} and B_{2}) with R_{1} ⊥ R_{2} (resp. B_{1} ⊥ B_{2}).

6 R. B. Howlett and J. Y. Shi

By Lemma 1.3 we see that if α ∈ B and β = ζα ∈ R for some ζ ∈ ℂ^{∗}, then B′ = (B \ {α}) ∪ {β} also forms a simple root system for R. 1.6. Let R be a root system with G the associated reflection group. For any α ∈ R, let R_{α} = { γ ∈ R | γ ∈ ℂα }, called a root line in R. By Property (iii) of a root system, we see that a root line of R is contained in a single G-orbit, and that the action of G on R induces a permutation on \overline{R}, the set of all root lines of R. It is easily seen that if α, β ∈ R then s_{β}(R_{α}) ≠ R_{α} unless β ∈ R_{α} or (β, α) = 0.

We have the following result concerning the root lines.

Lemma 1.7. Let R be a root system with G the associated reflection group. Then the order of the center Z(G) of G divides |R_{α}| for any α ∈ R.

Proof. Since G acts irreducibly on V, it follows by Schur’s Lemma that elements of Z(G) act as scalar multipliers. Hence R_{α} is a union of some Z(G)-orbits, each of which has the same cardinality |Z(G)|. So the result follows. □

1.8. In the subsequent sections, we always assume that G is an irreducible finite reflection group of rank greater than 1. Let R be the root system associated with G and B a simple system for R. Let S = { s_{α} | α ∈ B } be the set of the simple reflections associated with the simple system B; that is

s_{α}(v) = v + (e^{2πi/d_{α}} − 1)(v, α)α, (for all v ∈ V),

where d_{α} is the order of α (and also of s_{α}).

Let \overline{R} be the set of all root lines of R.

§2. Perfectness. 2.1. A sequence of root lines ξ : l_{1}, l_{2}, …, l_{r} (r > 1) in \overline{R} is called a circle if the following conditions hold.

Finite Reflection Groups 7

(i) l_{i} ≠ l_{i+1} for all i = 1, 2, …, r, with the convention that l_{r+1} = l_{1}. (ii) For any h with 1 ⩽ h ⩽ r, there exists some s_{h} ∈ S satisfying

s_{h}(l_{h}) = l_{h+1}. 2.2. Given two distinct root lines l and l′ such that l′ = s(l) for some s ∈ S, it is possible that there exists an s′ ∈ S such that s′(l) = l′ and s′ ≠ s. (This happens, for example, for the group G_{7} of Shephard and Todd [16]). Thus the sequence of simple reflections s_{1}, s_{2}, …, s_{r} associated with a root line circle ξ need not be unique. 2.3. We say that a root line circle ξ : l_{1}, l_{2}, …, l_{r} in \overline{R} is perfect (with respect to S), if for any sequence of simple reflections s_{1}, …, s_{r} satisfying s_{h}(l_{h}) = l_{h+1} for all h, we have s_{r}s_{r−1} ⋯ s_{1}(α) = α for each α ∈ l_{1}. Otherwise we say that the root line circle is imperfect. Remark 2.4. (1) If one root α in l_{1} satisfies the required condition that s_{r}s_{r−1} ⋯ s_{1}(α) = α, then so do all others; so when checking perfectness we need only consider a fixed α ∈ l_{1}. However, we do need to consider every possible sequence of simple reflections s_{1}, s_{2}, …, s_{r} associated with the root line circle, since, as explained in 2.2 above, it need not be unique.

(2) Perfectness is defined relative to a set S of simple reflections for G. In the sequel, S is always fixed, and so we shall not mention the dependence on S explicitly. 2.5. Let R = ⋃_{i∈I} R_{i} be the decomposition of R into a disjoint union of G-orbits R_{i}, and \overline{R} = ⋃_{i∈I} \overline{R}_{i} the corresponding decomposition of \overline{R}. We say that R and \overline{R} (resp. R_{i} and \overline{R}_{i}) are perfect if all the root line circles in \overline{R} (resp. \overline{R}_{i}) are perfect (relative to S). We say that G is perfect if its associated root system R is perfect.

Lemma 2.6. Let \overline{R}_{i} be a perfect G-orbit in \overline{R}. Suppose that we have two root line sequences ξ : l_{1}, …, l_{t} and η : l′_{1}, …, l′_{r} in \overline{R}_{i} with l_{1} = l′_{1} and 8 R. B. Howlett and J. Y. Shi

l_{t} = l′_{r}, and two simple reflection sequences s_{1}, …, s_{t−1} and s′_{1}, …, s′_{r−1} such that s_{h}(l_{h}) = l_{h+1} ≠ l_{h} and s′_{k}(l′_{k}) = l′_{k+1} ≠ l′_{k} for all h and k (with 1 ⩽ h < t and 1 ⩽ k < r). Take any root α ∈ l_{1}. Then the equation

s_{t−1}s_{t−2} ⋯ s_{2}s_{1}(α) = s′_{r−1}s′_{r−2} ⋯ s′_{2}s′_{1}(α)

is satisfied.

Proof. Let δ = s_{t−1}s_{t−2} ⋯ s_{2}s_{1}(α) and δ′ = s′_{r−1}s′_{r−2} ⋯ s′_{2}s′_{1}(α), and suppose that the result false. Then δ′ = cδ for some complex number c ≠ 1. For all h and k with 1 ⩽ h ⩽ t and 1 ⩽ k ⩽ r, put α_{h} = s_{h−1}s_{h−2} ⋯ s_{1}(α) and β_{k} = s′_{k−1}s′_{k−2} ⋯ s′_{1}(α). Thus α_{t} = δ and β_{r} = δ′, and α_{1} = β_{1} = α.

Define a new sequence of roots ζ′ : μ_{1} = δ, μ_{2}, …, μ_{p} = α from the sequence ζ : α_{1} = α, α_{2}, …, α_{t} = δ as follows. Start by reversing the order of ζ to get the sequence ζ^{−} : α_{t}, …, α_{2}, α_{1}; then, for each h (1 ⩽ h < t) such that the order m_{h} of s_{h} is greater than 2, insert the sequence of roots

s_{h}(α_{h+1}), s_{h}^{2}α_{h+1}), …, s_{h}^{m_{h}−2}(α_{h+1})

between α_{h+1} and α_{h} in ζ^{−}. This gives the desired sequence ζ′. Now let τ be the root line sequence l_{1}, l_{2}, …, l_{p+r−1} such that l_{h} contains μ_{h} for 1 ⩽ h ⩽ p, and l_{p+k} contains β_{k} for 1 ⩽ k < r. Then τ is an imperfect circle, contradicting our assumption. □

§3. Basic sections of root systems. As before, let B be a simple system for a root system R, let G be the associated reflection group and S = {s_{α} | α ∈ B} the set of simple reflections. 3.1. A subset R′ ⊂ R is called a section of R if it contains a unique root in each root line l ∈ \overline{R}. A section R′ of R is basic if, for every β ∈ B, the simple reflection s_{β} permutes the roots in the set R′ \ R_{β}. Equivalently, the

Finite Reflection Groups 9

section R′ is basic if s(γ) ∈ R′ whenever γ ∈ R′ and s is a simple reflection such that s(R_{γ}) ≠ R_{γ}.

Basicness, like perfectness, is defined relative to the set S, which we shall consider to be fixed.

A positive root system of a Coxeter group forms a basic section of its root system. Also, any root of a reflection group of rank 1 forms a basic section. 3.2. In general, a root system always contains a section, but need not contain a basic section. A finite reflection group G is called regular, relative to a set S of simple reflections, if its root system R contains a basic section.

The main goal of the present paper is to show that an irreducible finite reflection group G is regular if and only if it is either a Coxeter group or a group of rank 1.

By Lemma 2.6, we see that if all the root line circles of R are perfect then we can define a section R′ of R in the following way. Let R = ⋃_{i∈I} R_{i} be the decomposition of R into a disjoint union of G-orbits R_{i}, and \overline{R} = ⋃_{i∈I} \overline{R}_{i} the corresponding decomposition of the root line set \overline{R}. For any i ∈ I, fix a root α in R_{i}, and let l = R_{α} ∈ \overline{R}_{i} be the root line containing α. For any root line l′ ∈ \overline{R}_{i} there exists a sequence ξ : l_{1} = l, l_{2}, …, l_{r} = l′ in \overline{R}_{i} such that l_{h} ≠ l_{h+1} and s_{h}(l_{h}) = l_{h+1} for 1 ⩽ h < r and some s_{h} ∈ S. Define a root γ ∈ l′ by γ = s_{r−1}s_{r−2} ⋯ s_{1}(α). By Lemma 2.6, γ depends only on α and not on the choice of ξ; so each l′ contains a unique such γ. Let R^{(i)} be the set of all roots γ obtained from α in this way for l′ ∈ \overline{R}_{i}. Then the set R′ = ⋃_{i∈I} R^{(i)} is a section of R.

Lemma 3.3. The section R′ defined above is basic.

Proof. Let β ∈ B and γ ∈ R′ \ R_{β}. It suffices to prove that s_{β}(γ) ∈ R′. We may assume that (β, γ) ≠ 0, since otherwise s_{β}(γ) = γ ∈ R′.

10 R. B. Howlett and J. Y. Shi

Let R_{i} be the G-orbit of R in which γ lies, and let α be the fixed root in R_{i} used in defining R′. Then by the definition of R′ there exist root lines l_{1} = R_{α}, l_{2}, …, l_{r} = R_{γ} and simple reflections s_{1}, …, s_{r−1} such that l_{h} ≠ l_{h+1} and s_{h}(l_{h}) = l_{h+1} for 1 ⩽ h < r, and s_{r−1} ⋯ s_{1}(α) = γ. Now as β ∉ ℂγ and (β, γ) ≠ 0, it follows that s_{β}(l_{r}) = s_{β}(R_{γ}) ≠ R_{γ}, and so if we define l_{r+1} = s_{β}(l_{r}) then the conditions l_{h} ≠ l_{h+1} and s_{h}(l_{h}) = l_{h+1} hold for 1 ⩽ h < r + 1. Hence s_{β}(γ) = s_{r}s_{r−1} ⋯ s_{1}(α) is in R′, as required. □

Lemma 3.3 immediately yields the “if” part of the following Theorem.

Theorem 3.4. An irreducible finite reflection group G is regular if and only if all the root line circles in \overline{R} are perfect.

Proof. By Lemma 3.3, it suffices to show the implication ⟹. Let R′ be a basic section of R. Let ξ : l_{1}, …, l_{r} be a root line circle of \overline{R} and s_{1}, …, s_{r} simple reflections with s_{h}(l_{h}) = l_{h+1} for 1 ⩽ h ⩽ r. Choose a root α_{1} ∈ l_{1}. By Remark 2.4 (1) we can assume without loss of generality that α_{1} ∈ R′. Proceeding inductively, define α_{j+1} = s_{j}(α_{j}) for 1 ⩽ j ⩽ r. Since s_{j}(l_{j}) ≠ l_{j} it follows from the definition of a basic section that α_{j} ∈ R′ for all j. Since α_{r+1} ∈ l_{1} ∩ R′ and since α_{1} is the unique root in l_{1} ∩ R′, this implies that α_{r+1} = α_{1}, and so the root line circle ξ is perfect. □

§4. The main result. In this section, we assume that G is an irreducible finite regular reflection group of rank > 1 with S a set of simple reflections. Let (R, o_{G}) be the root system associated with G and let B be a simple root system for R contained in a basic section R′ of R. Let \overline{R} be the root line set of R.

Lemma 4.1. For any α ∈ R, the root line R_{α} has cardinality o_{G}(α).

Proof. Without loss of generality, we may assume that α ∈ R′. Since s_{α}

Finite Reflection Groups 11

acts on R_{α} by multiplication by the scalar ζ = e^{2πi/o(α)}, it follows that R_{α} is a union of ⟨s_{α}⟩-orbits, each orbit having the same cardinality o(α). We shall show that R_{α} actually consists of a single ⟨s_{α}⟩-orbit, and hence has cardinality o(α).

Let Z be the subgroup of ℂ^{∗} generated by ζ, and let γ ∈ R_{α} be arbitrary. Since R_{α} = ℂα ∩ Gα, there exists an element y ∈ G with γ = y(α), and we may choose s_{1}, s_{2}, …, s_{r} ∈ S such that y = s_{r}s_{r−1} ⋯ s_{1}. Put α_{0} = α, and for 1 ⩽ k ⩽ r, put α_{k} = s_{k}s_{k−1} ⋯ s_{1}(α) and let l_{k} be the root line containing α_{k}. We use induction on k to show that for each k, there exists an α′_{k} ∈ R′ such that α_{k} ∈ Zα′_{k}.

The case k = 0 is trivial since α_{0} = α ∈ R′. So assume that k > 1 and α_{k−1} = ζ^{m}α′_{k−1} for some α′_{k−1} ∈ R′ and m ∈ ℤ. If s_{k}(l_{k−1}) ≠ l_{k−1} then by the definition of a basic section the root s_{k}(α_{k−1}) is in R′; so defining α′_{k} = s_{k}(α′_{k−1}), we conclude that α_{k} = s_{k}(α_{k−1}) = ζ^{m}α′_{k}, as required. So we may assume that s_{k}(l_{k−1}) = l_{k−1}, and hence either β ∈ l_{k−1} or else (β, α_{k−1}) = 0, where β is the simple root such that s_{β} = s_{k}. In the latter case α_{k} = s_{β}(α_{k−1}) = α_{k−1}, and defining α′_{k} = α′_{k−1} ∈ R′ gives α_{k} ∈ Zα′_{k} as required. On the other hand, if β ∈ l_{k−1}, then by Lemma 1.3, we have o(β) = o_{G}(α_{k−1}) = o_{G}(α), since α_{k−1} and α are in the same G-orbit. Hence s_{β} acts on l_{k−1} by multiplication by ζ; so α_{k} = s_{β}(α_{k−1}) = ζα_{k−1} = ζ^{m+1}α′_{k}, where α′_{k} = α′_{k−1} ∈ R′, and this completes the induction.

Since γ = α_{r}, we conclude from the above that γ ∈ Zα′_{r}, where α′_{r} ∈ R′∩l_{r}. But l_{r} = R_{γ} = R_{α}, and α ∈ R′ by hypothesis. Since R′ is a section, it follows that α′_{r} = α, and thus γ ∈ Zα. But γ was chosen as an arbitrary element of Rα, and so we conclude that Rα = Zα, as required. □

Lemma 4.2. (1) For any α ∈ R, we have |R_{α}| = o(s) for some s ∈ S; (2) |Z(G)| is a divisor of o(s) for every s ∈ S.

12 R. B. Howlett and J. Y. Shi

Proof. Since R = GB, there is an element β ∈ B with α = g(β) for some g ∈ G. By Lemmas 4.1 and 1.3, we have |R_{α}| = o_{G}(α) = o_{G}(β) = o(s_{β}). So (1) follows. Then (2) follows from (1), Lemma 1.7 and R = GB. □

Proposition 4.3. Let G be an irreducible finite complex reflection group of rank > 1 with R the associated root system. If |Z(G)| is a divisor of o(s) for every s ∈ S, then G must be one of the following groups (in the notation of Shephard and Todd, see [4; 16]):

(1) the primitive complex reflection groups G_{8}, G_{12}, G_{24}, G_{25}, G_{33};

(2) the imprimitive complex reflection groups G(m, m, n) (with m > 2 and gcd(m, n) ∈ {1, 2}) and G(2p, p, n) (with p > 1 and gcd(p, n) = 1).

Proof. This follows from [3, Tables 1-4], using the fact that the order of Z(G) is the greatest common divisor of the degrees of G. □

4.4. We shall show that none of the groups listed in Proposition 4.3 is regular. By Theorem 3.4, it suffices to find in each case a sequence of roots γ_{0}, γ_{1}, …, γ_{t} in the root system R associated with G such that the root lines R_{γ_{0}}, R_{γ_{1}}, …, R_{γ_{t}} form an imperfect circle.

Let ϵ_{1}, ϵ_{2}, … be an orthonormal basis in the hermitian space V. We shall give a simple root system B = {α_{1}, α_{2}, … } associated with G, expressing the α_{i}’s in terms of the ϵ_{i}’s. The order o_{G}(α_{i}) of α_{i} will be 2 unless otherwise specified. We specify roots of G by giving their coordinates relative to B if B is linearly independent, or relative to some maximal linearly independent subset of B if B is linearly dependent. We use the notation i = \sqrt{−1} ∈ ℂ. 4.5. G = G_{8}: Let B = \{α_{1}, α_{2}\} with α_{1} = ϵ_{1} and α_{2} = \frac{1+i}{2}(ϵ_{2} − ϵ_{1}), and o_{G}(α_{1}) = o_{G}(α_{2}) = 4. Then γ_{0} = (0, 1), γ_{1} = (1, 1), γ_{2} = (1, 0) and γ_{3} = (1, −i) are four roots of G. We have

γ_{0} ⟼^{s_{1}} γ_{1} ⟼^{s_{2}} γ_{2} ⟼^{s_{2}} γ_{3} ⟼^{s_{1}} −iγ_{0},

Finite Reflection Groups 13

where the notation a ⟼^{s} b means s(a) = b. 4.6. G = G_{12}: Let B = \{α_{1}, α_{2}, α_{3}\} with α_{1} = ϵ_{1}, α_{2} = \frac{−1+i\sqrt{2}}{2}ϵ_{1}+\frac{\sqrt{3}−i\sqrt{6}}{6}ϵ_{2} and α_{3} = −\frac{i\sqrt{2}}{2}ϵ_{1} + \frac{2\sqrt{3}+i\sqrt{6}}{6}ϵ_{2} = α_{1} + i\sqrt{2}α_{2}. Using coordinates relative to α_{1} and α_{2}, the four roots given by γ_{0} = (0, 1), γ_{1} = (1 + i\sqrt{2}, −1 + i\sqrt{2}), γ_{2} = (i\sqrt{2}, −1 + i\sqrt{2}) and γ_{3} = (−1 + i\sqrt{2}, −1) satisfy

γ_{0} ⟼^{s_{3}} γ_{1} ⟼^{s_{1}} γ_{2} ⟼^{s_{3}} γ_{3} ⟼^{s_{1}} −γ_{0}.

4.7. G = G_{24}: Let B = \{α_{1}, α_{2}, α_{3}\} with α_{1} = ϵ_{2}, α_{2} = \frac{\overline{λ}}{2}(ϵ_{2} + ϵ_{3}) and α_{3} = −\frac{1}{2}(ϵ_{1} + ϵ_{2} − λϵ_{3}), where λ = \frac{1}{2}(1 + i\sqrt{7}). Then

γ_{0} = (1, 0, 0)  γ_{1} = (1, 0, 1)  γ_{2} = (0, 0, 1)
γ_{3} = (0, 1, 1)  γ_{4} = (0, 1, 0)  γ_{5} = (−\overline{λ}, 1, 0)
γ_{6} = (−\overline{λ}, 1, λ)  γ_{7} = (−\overline{λ}, 1 + λ, λ)  γ_{8} = (−1 − \overline{λ}, 1 + λ, λ)
γ_{9} = (−1 − \overline{λ}, 1 + λ, −\overline{λ}) γ_{10} = (−1 − \overline{λ}, λ, −\overline{λ}) γ_{11} = (−1, λ, −\overline{λ})
γ_{12} = (−1, λ, 0)

are thirteen roots of G. We have

γ_{0} ⟼^{s_{3}} γ_{1} ⟼^{s_{1}} γ_{2} ⟼^{s_{2}} γ_{3} ⟼^{s_{3}} γ_{4} ⟼^{s_{1}} γ_{5} ⟼^{s_{3}} γ_{6} ⟼^{s_{2}} γ_{7}

⟼^{s_{1}} γ_{8} ⟼^{s_{3}} γ_{9} ⟼^{s_{2}} γ_{10} ⟼^{s_{1}} γ_{11} ⟼^{s_{3}} γ_{12} ⟼^{s_{2}} −γ_{0}.

4.8. G = G_{25}: Let B = \{α_{1}, α_{2}, α_{3}\} with α_{1} = \overline{ω}ϵ_{3}, α_{2} = (i/\sqrt{3})(ϵ_{1}+ϵ_{2}+ϵ_{3}), α_{3} = \overline{ω}ϵ_{2}, and o_{G}(α_{j}) = 3 for each j ∈ \{1, 2, 3\}, where ω is a primitive cube root of unity. Then γ_{0} = (1, 1, 0), γ_{1} = (−\overline{ω}, 1, 0) are two roots of G, and

γ_{0} ⟼^{s_{1}} γ_{1} ⟼^{s_{2}} −\overline{ω}γ_{0}.

4.9. G = G_{33}: Let B = {α_{j} | 1 ⩽ j ⩽ 5} with α_{1} = 2^{−1/2}ω(ϵ_{5} + ϵ_{6}), α_{2} = −2^{−3/2}ω(−ϵ_{1} + (1 + 2ω)ϵ_{2} + ϵ_{3} + ϵ_{4} + ϵ_{5} + ϵ_{6}), α_{3} = 2^{−1/2}(ϵ_{1} − ϵ_{2}), α_{4} = 2^{−1/2}(ϵ_{2} − ϵ_{3}) and α_{5} = 2^{−1/2}(ϵ_{3} − ϵ_{4}). Then γ_{0} = (0, 0, 0, 1, 0), 14 R. B. Howlett and J. Y. Shi

γ_{1} = (0, 0, 1, 1, 0), γ_{2} = (0, 0, 1, 0, 0), γ_{3} = (0, 1, 1, 0, 0), γ_{4} = (0, 1, 0, 0, 0) and γ_{5} = (0, 1, 0, \overline{ω}, 0) are six roots of G such that

γ_{0} ⟼^{s_{3}} γ_{1} ⟼^{s_{4}} γ_{2} ⟼^{s_{2}} γ_{3} ⟼^{s_{3}} γ_{4} ⟼^{s_{4}} γ_{5} ⟼^{s_{2}} \overline{ω}γ_{0}.

We see that in each of the above five groups, the corresponding root lines R_{γ_{0}}, R_{γ_{1}}, … form an imperfect circle. 4.10. It remains to find imperfect root line circles for the imprimitive complex reflection groups listed in Proposition 4.3 (2). We shall actually do this for all the groups G(m, m, n) with m > 2 and G(2p, p, n) with p > 1, without the restrictions gcd(m, n) ∈ {1, 2} and gcd(p, n) = 1.

For each m > 2, let ζ_{m} = e^{2πi/m}. The group G(m, m, n) has a simple root system B = \{α_{1}, α_{2}, …, α_{n}\} given by α_{i} = \frac{1}{\sqrt{2}}(ϵ_{i} − ϵ_{i+1}) for 1 ⩽ i < n and α_{n} = \frac{1}{\sqrt{2}}(ϵ_{1} − ζ_{m}ϵ_{2}). The root system R generated by B consists of all the vectors ±(ζ_{m}^{k}ϵ_{i}−ζ_{m}^{l}ϵ_{j}), where i, j are distinct elements of {1, 2, …, n} and k, l are integers. Then γ_{0} = \frac{1}{\sqrt{2}}(ϵ_{2} −ϵ_{3}), γ_{1} = \frac{1}{\sqrt{2}}(\overline{ζ}_{m}ϵ_{1} −ϵ_{3}), γ_{2} = \frac{1}{\sqrt{2}}(\overline{ζ}_{m}ϵ_{2} −ϵ_{3}), γ_{3} = \frac{1}{\sqrt{2}}(\overline{ζ}_{m}ϵ_{3} − ϵ_{2}) and γ_{4} = \begin{matrix}\overline{ζ} & _{m} \\ √ & \overline{2} \end{matrix} (ϵ_{3} − ϵ_{1}) are five roots of G(m, m, n). Since ζ_{m} ≠ ±1, they belong to five different root lines. Now

γ_{0} ⟼^{s_{n}} γ_{1} ⟼^{s_{1}} γ_{2} ⟼^{s_{2}} γ_{3} ⟼^{s_{n}} γ_{4} ⟼^{s_{1}} −\overline{ζ}_{m}γ_{0},

whence R_{γ_{0}}, R_{γ_{1}}, R_{γ_{2}}, R_{γ_{3}}, R_{γ_{4}} constitute an imperfect root line circle. 4.11. The group G(2p, p, n) has a simple root system B ={α_{1}, α_{2}, …, α_{n}, α′_{n}} with α_{j} as in 4.10 for 1 ⩽ j < n, while α_{n} = \frac{1}{\sqrt{2}}(ϵ_{1} −ζ_{2p}ϵ_{2}) and α′_{n} = ϵ_{1}. The root system generated by B is R(2p, p, n) = {±ϵ_{k} | 1 ⩽ k ⩽ n}∪R(2p, 2p, n), where R(2p, 2p, n) is the root system of G(2p, 2p, n) as described in 4.10. In particular, G(2p, p, n) contains G(2p, 2p, n) as a standard parabolic subgroup (see 1.5); so the existence of an imperfect root line circle in R(2p, p, n) follows from the existence of one in R(2p, 2p, n), as established in 4.10.

We are now ready to prove our main result.

Finite Reflection Groups 15

Theorem 4.12. An irreducible finite reflection group G is regular if and

only if G is either a Coxeter group or has rank 1.

Proof. The implication ⇐ is well known (see 3.1); so it suffices to show that if

G has rank > 1 then G is not regular. By Lemma 4.2 (2) and Proposition 4.3,

if G is regular, then it must be one of the groups listed in Proposition 4.3.

The results in 4.5–4.11 show that each of these has at least one imperfect

root line circle in its root system, and so is not regular by Theorem 3.4. □

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