On Regularity of Finite Reflection Groups

We define a concept of “regularity” for finite unitary reflection groups, and show that an irreducible finite unitary reflection group of rank greater than 1 is regular if and only if it is a Coxeter group. Hence we get a characterization of Coxeter groups among all the irreducible finite reflection groups of rank greater than one. The irreducible finite unitary reflection groups were classified by Shephard and Todd ([12]) and by Cohen ([5]). They have been studied extensively by many people since then (see [2; 3; 4; 6; 7; 8; 9; 10; 11; 13]). 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 consider a property, which we call regularity, of a reflection group, defined by the existence of a basic section of an associated root system R (see 3.1). We show that an irreducible finite reflection group G of rank greater than 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 and only if G is a Coxeter group. Thus we get a characterization of Coxeter groups among all the finite reflection groups. The second named author is grateful for the financial support of the Australian Research Council and for the hospitality of the University of Sydney. He is also supported partly by the National Science Foundation of China and the Science Foundation of the University Doctorial Program of CNEC Typeset by AMS-TEX 1 2 R. B. Howlett and J. Y. Shi Let G be an irreducible finite reflection group which acts irreducibly on a unitary space V , and let S be a simple reflection set for G (see 1.5). The strategy for proving our main result is as follows. It suffices to show that if G is complex and has rank greater than 1 then it is not regular. 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 a root system R of G (see 1.6). We also show in Theorem 3.4 that G is regular if and only if all root line circles in R (see 2.1) are perfect. Suppose that G is regular and of rank greater than 1. Then we further show in Lemma 4.2 (1) that for any root α in R, the cardinality of the root line Rα is equal to the order of some s ∈ S. We show in Lemma 4.2 (2) that |Z(G)| divides the orders o(s) of all s ∈ S. We also show in Lemma 4.2 (3) that if the cardinality of each root line of R is 2, then G is a Coxeter group. By the classification of the irreducible finite unitary reflection groups, there are only three complex groups of rank greater than 1 satisfying both conditions (1), (2) of Lemma 4.2, namely, the groups G8, G12 and G24 (in the notation of Shephard and Todd [12]). But the groups G12 and G24 are not regular by Lemma 4.2 (3), and we show that G8 is not regular by exhibiting an an imperfect root line circle in its root system. 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 4.4, 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 [5]. Finite Reflection Groups 3 1.1. Let V be a complex vector space of dimension n. A reflection on V is a linear transformation on V of finite order with exactly n−1 eigenvalues equal to 1. A reflection group G on V is a finite group generated by reflections on 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 to be irreducible if V has no nonzero proper G-invariant subspaces. In the present paper we shall always assume that G is irreducible and acts irreducibly on V . Call the dimension of V the rank of G. A reflection group G on V is called a real group or a Coxeter group if there is a G-invariant R-subspace V0 of V such that the canonical map C⊗R V0 → 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.) Since G is finite, there exists a unitary inner product ( , ) on V invariant under G. From now on we assume that such an inner product is fixed. 1.2. A root of a reflection on 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 on 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 = sa,ζ , where sa,ζ is defined by (1.2.1) sa,ζ(v) = v + (ζ − 1)(v, a)a for all v ∈ V . We also write sa,d for sa,ζ if ζ = e. Note that a can be chosen to be any root of s of length 1, and ζ is the nontrivial eigenvalue of s. 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 oG(v) to be the order of the (necessarily cyclic) group that consists of the identity and the reflections in G which have v as a root. (This group is GW = { g ∈ G | gu = u for all u ∈W }, where W = v⊥.) Thus oG(v) > 1 if and only if v is a root of G. If a is a root of G, then oG(a) will be called the order of a (with respect to G). We shall denote oG(a) simply by o(a) when G is clear from the context. 4 R. B. Howlett and J. Y. Shi 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. We have o(gv) = o(v) = o(cv) for all v ∈ V , g ∈ G and c ∈ C∗, where C∗ := C \ {0}. Proof. Let W = v⊥. Then u ∈W ⇐⇒ (u, v) = 0⇐⇒ (gu, gv) = 0⇐⇒ gu ∈ (gv)⊥. This implies that gW = (gv)⊥. Now we have h ∈ GgW ⇐⇒ h(gu) = gu for all u ∈W , ⇐⇒ (g−1hg)u = u for all u ∈W , ⇐⇒ g−1hg ∈ GW , ⇐⇒ h ∈ GW . So we get GgW = GW and hence o(gv) = |GgW | = |GW | = |GW | = 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 sa,f(a)R = R and f(sa,f(a)(b)) = f(b) for all a, b ∈ R; (iii) the group G generated by { sa,f(a) | a ∈ R } is a (finite) reflection group, and for all a ∈ R and c ∈ C, ca ∈ R⇐⇒ ca ∈ Ga. The group G is called the reflection group associated with the root system (R, f). We have oG(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: Finite Reflection Groups 5 (i) for all a, b ∈ B, we have |(a, b)| = 1⇐⇒ a = b; (ii) the group G generated by S = { sa,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) the group G cannot be generated by fewer than |B| reflections. We call the elements of S simple reflections. 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 requireB to be linearly independent. IfB is linearly independent, then condition (iv) holds automatically. The above definition of simple system is considerably weaker than the usual definition for Coxeter groups; in particular, it is not true that if B1 and B2 are simple systems for the same root system R then there is an element g ∈ G with gB1 = B2. 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 R1 and R2 (resp. B1 and B2) with R1 ⊥ R2 (resp. B1 ⊥ B2). By Lemma 1.3 we see that if α ∈ B and β ∈ Cα∩R, then B′ = (B \ {α})∪ {β} also forms a simple root system for R. 1.6. Let R be a root system in V with G the associated reflection group. For any α ∈ R, let Rα = Cα ∩R, 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 action on R, the set of all root lines of R. It is easily seen that if α, β ∈ R then sβ(Rα) 6= Rα unless β ∈ Rα 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 |Rα| 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 Rα is a union of some Z(G)-orbits, each of which has the same cardinality |Z(G)|. So the result follows. 6 R. B. Howlett and J. Y. Shi 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 an associated root system and B a simple system for R. Let S = { sα | α ∈ B } be the set of the simple reflections associated with the simple system B; that is sα(v) = v + (eα − 1)(v, α)α (for all v ∈ V ) where dα is the order of α (and also of sα). 1.9. Let (R, f) be a root system with G the associated reflection group. Then sa,f(a) is a reflection in G for any a ∈ R and any integer k with 1 6 k < f(a). Conversely, every reflection in G has such a form. For a, b ∈ R, if 1 6 h < f(a) and 1 6 k < f(b) then the reflections sa,f(a) and s k b,f(b) are G-conjugate if and only if a and b are in the same G-orbit and h = k. It is possible that there is another root system (R′, f ′) in V with the same associated reflection group G. In that case, we can find, for any a ∈ R, a root a′ ∈ R′ such that sa,f(a) = sa′,f ′(a′). This implies that a′ ∈ Ca. Similarly, for any a′ ∈ R′, we can find some a ∈ R with a ∈ Ca′. According to the condition (iii) of a root system, we see that the cardinality of a root line Ra in R is equal to that of the root line Ra′ in R′. In particular, when R consists of a single G-orbit, (R, f) is determined by G up to a scalar factor. These facts will be used in Section 4 in the proof of our