Complex Reflection Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 10 January 2014

Section IV

Let V be an dimensional complex vector space and let P denote the ring of polynomial functions on V. If GGL(V), then G acts naturally on P and the elements of P invariant under G form a subring I(P) of P. In [STo1954] it is shown that G is a finite group generated by reflections if and only if I(P) is generated by algebraically independent homogeneous polynomials f1,,f. Let di denote the degree of fi. The integers d1,,d are uniquely determined by G and the numbers d1-1,,d-1 are called the exponents of group G. From [STo1954] we have |G|=i=1di and that the number of reflections in G is i=1(di-1). In [STo1954], Shephard and Todd further show that if G may be generated by reflections then one may choose generating reflections R1,,R so that the eigenvalues of the product R1,,R are γd1-1,,γd-1 where γ=e2πi/h and d=h is the order of R1R. In their argument there is no general algorithm for choosing such generators. However, if G=θ(W(Γ)), for some connected Γ𝒞+ then there is a natural choice for such generators.

Theorem 6. Let Γ𝒞+ be connected. As usual we let r1,,r be the generators for W(Γ) and we let Si=θ(ri). Put Ri=Si-1 and let Π be any permutation of {1,,}. Then

(i) The conjugacy class of rΠ(1)··rΠ() does not depend on Π.
(ii) The eigenvalues of R1··R are γd1-1,,γd-1 where γ=e2πi/h and h is the order of R1··R.
(iii) d=h.


By Theorem 2 Γ is a tree and thus (i) is given in [Bou1968] (Lemma 1, Chapter V, section 6). For a graph Γ with just one vertex labelled with the integer p we see from [STo1954] that d1=p and thus (ii) and (iii) are obvious. Now if Γ satisfies pi=2, 1i, so W(Γ) is a Coxeter group, (ii) and (iii) are given by a general argument in [Bou1968] (Proposition 3, Chapter V, section 6). If Γ𝒞2+, say Γ is p1[q]p2, Coxeter observes [Cox1962] that d1=2h/q and d2=h. He also obtains γ and γ1+h-2hq as the eigenvalues of S1tS2t and thus we have γ2hq-1 and γh-1 as the eigenvalues of R1R2.

We now consult List 1 of Theorem 2 to determine which graphs still remain and we will verify (ii) and (iii) for these groups in a case by case manner.

Suppose Γ is 3[3]3[3]3. By comparing [3] with [2] we see that d1=6, d2=9, d3=12. Now the matrix for R1R2R3 in the basis {v1,v2,v3} is ( 00-ω α0ω2α 0αω2 ) where ω=e2πi/3 and α=1-ω23.

The characteristic equation is thus λ3-ω2λ2+ωλ-1=0 which has roots γ5, γ8, γ11 where γ=e2πi/12.

For the graph Γ 3[3]3[4]2 we find from [3] and [2] that the degrees are d1=6, d2=12, d3=18. We compute that the matrix for R1R2R3 in the basis {v1,v2,v3} is ( 0αBω2 α1-αB 0B-1 ) where α=1-ω23 and B=314. Thus the characteristic equation is λ3+ω=0 and the eigenvalues of R1R2R3 are γ5, γ11, γ17 where γ=e2πi/18.

If Γ is 3[3]3[3]3[3]3 we find again from [3] and [2] that d1=12, d2=18, d3=24, d4=30. This time the matrix for R1R2R3R4 in the basis {v1,v2,v3,v4} is ( 000-ωα α00-ω 0α0ω2α 00αω2 ) where ω=e2πi/3 and α=1-ω23. The characteristic equation is thus λ4-ω2λ3+ωλ2-λ+ω2=0 which has roots γ11, γ17, γ23, γ29 where γ=e2πi/30.

We finally consider the infinite family of graphs Bp. Here, comparing [3] and [2] yields that the degrees are given by dk=kp (1k). Letting α={2sinπ/p}-12 we define a new basis {x1,,x} by xi=αv1+v2+v3++vi. Then S1(x1)=εx1 (ε=e2πi/p) S1(xj)=xj forji. Further if 2k, Sk interchanges xk-1 and xk and fixes every other xj. Hence the matrix for R1R2··R. in this basis is ( 0ε-1 10 10 1 0 10 ) . The characteristic equation is thus λ-ε-1=0. Hence h=p and letting γ=e2πi/h the roots are γkp-1 1k.

This completes the verification.

Let Γ𝒞+ be connected. Put S=S1··S, let h denote the order of S and let γ=e2πi/h. We define the integers n1,,n (0n1nh-1) and m1,,m (0m1m2mh-1) by requiring that γnj, 1j are the eigenvalues for S and γmj, 1j, are the eigenvalues for S-1. Thus, using Theorem 6, mi=di-1. In order to give a general argument for Theorem 6 along the same lines as that given in [Bou1968] for the case where W(Γ) is a Coxeter group one needs to know three things:

(a) n1=1
(b) There is an eigenvector corresponding to the eigenvalue γ of S which does not lie in the reflecting hyperplane of any reflection in θ(W(Γ)).
(c) The number of reflections in θ(W(Γ)) is i=1mi.
We have not been able to supply a general argument for these three. Note that by contrast to (c) one can give a case free argument for the assertion that the number of reflections in θ(W(Γ)) is i=1(di-1) [STo1954, p.289,290].

The quantity k=1ωk arises also in another setting. We have Det(S-1)=e2πihk=1mk. On the other hand, Det(S-1)= k=1Det (Sk-1)= e2πik=1pk-1pk . Thus, (7) 1hk=1 mkk=1 pk-1pk mod. Similarly, by computing Det(S) in two ways we obtain (8) 1hk=1 nkk=1 1pkmod. In case W(Γ) is a Coxeter group, all pi=2, S is conjugate to S-1, and both (7) and (8) become (9) k=1mk k=1nk h2mod. These congruences are in fact equalities [Bou1968, p.118] and this leads one to suspect that (7) and (8) are also equalities. Since mk+n-k+1=h equality holds in (7) if and only if it holds in (8).

Theorem 7. Let Γ𝒞+ be connected. With m1,,m and h defined as above we have 1hk=1mk =k=1 pk-1pk.


We may assume the pk are not all 2. If =1 the statement is obvious. If =2, Γ is p1[q]p2. From [Cox1962] we have that m1=2hq-1, m2=h-1, and h=2p1p2q(p1+p2)q-p1p2(q-2). Thus we have 1h(m1+m2) = 2q-1h+1-1h =2q+1-2h = 2q+1- p1+p2p1p2 +1-2q = (1-1p1)+1- 1p2 = p1-1p1+ p2-1p2. If 3 we have just computed the numbers m1,,m and h in the verification of Theorem 6. It thus becomes a trivial arithmetic task to verify the desired result for the remaining cases.

Corollary 7. Let Γ𝒞+ be connected. With h defined as above we have that the number of reflections in θ(W(Γ)) is hk=1pk-1pk.


Since mi=di-1, and i=1(di-1) is the number of reflections in θ(W(Γ)) this follows immediately from Theorem 7.

Coxeter [Cox1974, p.153] credits McMullen with the observation that for a connected graph Γ𝒞+ the number of reflecting hyperplanes in V corresponding to reflections in θ(W(Γ)) is hk=11pk. Now McMullen's observation does not seem to be a consequence of Corollary 7 nor does the corollary appear to follow from the observation. However, taken together we get the marvelous fact

Corollary 8. Let Γ𝒞+ be connected. If h is defined as above then the number of reflecting hyperplanes in V plus the number of reflections in θ(W(Γ)) is h.

Let Γ𝒞 and let I denote the set of vertices of Γ. For JI we denote by Γ(J) the subgraph of Γ obtained by deleting from Γ all those vertices in I\J and the edges connected to those vertices. Also we put W(J)=W(Γ(J)). Here we agree that if J=ϕ, then W(J)=1. Finally we write (-1)J for (-1)|J|.

Proposition 10. Let Γ𝒞+ be connected. Put W=W(Γ) and let I denote the vertex set of Γ. Put mi=di-1 where d1-1d-1 are the exponents of θ(W). Then (10) JI(-1)J |W:W(J)|= m1.


If W is a Coxeter group, m1=1 and thus (10) occurs in [Sol1966, p.378] where Solomon credits it to Witt.

If Γ has just one vertex labelled with the integer p, we have |W|=p and m1=p-1 so (10) is trivial.

Suppose next that Γ has two vertices; say Γ is p1[q]p2. Letting h denote the order of S1S2 we know from Theorem 6 that h=m2+1 and thus from Theorem 7 that 1-1p1-1p2=m1-1m2+1. Using this together with the previously mentioned fact that |W|=(m1+1)(m2+1) we compute JI (-1)J |W:W(J)| = |W| (1-1p1-1p2) +1 = (m1+1) (m2+1) (m1-1) m2+1 +1 = m12.

Now ignoring the infinite family Bp for a moment and using the table in [Cox1967] in conjunction with that in [STo1954] the verification of (10) for the other three non-Coxeter groups is a trivial arithmetic task.

Finally suppose Γ is the graph Bp p 4 Consulting [STo1954] and [Cox1967] we see that |W|=p! and m1=p-1. We number the vertices from left to right and denote the ith vertex by ai. We must show that JI(-1)J p!/|W(J)| =(p-1). Let S be the set of all subsets of the vertex set I={a1,,a}. Write S=k=0Sk a disjoint union where we put a subset JI in Sk if {a1,,ak}J and ak+1J. Let Ik={ak+2,,a} for 0k-2 and put I-1=I=ϕ. If JSk then J={a1,,ak}H where HIk and thus |W(J)|= pkk!|W(H)| (Here of course |W(ϕ)|=1).

Hence JI(-1)J |W(J)|-1= k=0 (-1)kpkk! HIk (-1)H |W(H)|-1 But W(Ik) is the symmetric group on -k letters, a Coxeter group, so (10) implies HIk(-1)H |W(H)|-1= 1(-k)! Thus JI (-1)J |W:W(J)| = k=0 (-1)k !k!(-k)! p-k = (p-1).

In light of the character formula which is proved in [Sol1966] (Theorem 2, p. 379) one is tempted to view (10) as the result of evaluating an equation involving an alternating sum of induced characters at the identity.

Let Γ𝒞+ be connected and put W=W(Γ). For wW we denote by μ(w) the multiplicity of 1 as an eigenvalue of θ(w) and we put k(w)=-μ(w).

Define Ψ:W by Ψ(w)= JI (-1)J 1W(J)W(w) where 1W(J)W is the character of W induced from the principal character of W(J). We offer the following

Conjecture: Let wW. Then (11) Ψ(w)= (-1)k(w) m1μ(w) (Here m1=d1-1, the smallest exponent of θ(W).)

Note that if W is a Coxeter group, m1=1. The non real eigenvalues of θ(w) occur in conjugate pairs and the real eigenvalues are ±1. Thus (-1)k(w)=Det(w) and hence, in this case, our conjecture is just Theorem 2 of [Sol1966] with χ=1W.

If the graph Γ has only one vertex (11) is obvious.

If Γ has just two vertices we will give an argument verifying (11) but we need to do some preliminary work first.

Lemma 4. Suppose G is a linear group. Let ZG be the subgroup of all those scalar matrices in G. Put G=G/Z and denote the natural map GG by gg for gG. Define G(g) = {xG:gx=g}and G(g) = {xG:gx=±g}. Then

(a) If gG satisfies tr(g)0, then G(g) =G(g)
(b) If G consists of 2×2 matrices, and gG satisfies tr(g)=0 then G(g)=G(g)


For both (a) and (b) it is clear that the right side is included in the left. So suppose xG(g). Hence gx=λg some λZ. Taking traces on both sides of this equation we have tr(g)=λtr(g), so tr(g)0 implies λ=1. Thus gx=g and xG(g), yielding (a). In the situation of (b) we take determinants on both sides of the equation gx=λg to obtain Det(g)=λ2Det(g). Thus λ2=1, so gx=±g and xG(g).

Corollary 9. Suppose Γ𝒞2+ is connected; say Γ is p1[q]p2. As usual we denote the generators for W(Γ) by r1, r2 and let G=θ(W(Γ)) with Si=θ(ri). Then W(ri)= ri×𝒵(W).


Suppose pi2. Since G is irreducible 𝒵(G) consists of scalar matrices. Put G=G/𝒵(G). Since Si is a reflection Si has order pi. From [STo1954] we know that G is the alternating group on four letters, the symmetric group on four letters, or the alternating group on five letters, and any non identity element of order different from two in any of these groups is self centralizing. Hence G(Si)=Si. Now pi2 further forces tr(Si)0, and applying Lemma 4(a) we have G(Si)=Si. Thus G(Si)=Si×𝒵(G).

Now suppose pi=2. Consulting List 1 of Theorem 2 we see Γ must be one of 3[6]2, 4[6]2, 3[8]2, 5[6]2, 3[10]2. From [STo1954] we have 2||𝒵(G)| so that -I𝒵(G). Hence -S2 is a reflection in G. It follows immediately from Theorem 5 that -S2 is conjugate to S2 for all of the above graphs with the possible exception of 4[6]2 where we must rule out the possibility that -S2 is conjugate to S12.

In the group 4[6]2, S1S2 has order 24 and 𝒵(G) is generated by (S1S2)3, an element of order 8. Hence -I=(S1S2)12. If λ is a linear character of G, λ(S1)=iα and λ(S2)=(-1)β and thus λ(S1S2)12=1. Thus -I=(S1S2)12G, the commutator subgroup of G. So, -S2 and S2 have the same images in G/G; but a glance at the proof of Lemma 3 reveals that S2 and S12 have distinct images in G/G. Hence, -S2 is not conjugate to S12.

So using the notation of Lemma 4 we have G(S2)> G(S2). In fact, we have G(S2): G(S2) =2. In light of Lemma 4(b) we see that to show G(S2)=S2×𝒵(G) it suffices to show (12) |G(S2)| =4. If G is the alternating group on tour or five letters (12) is obvaous. So we assume G is the symmetric group on four letters. Consulting [STo1954] we see that Γ is 3[8]2 or 4[6]2. If S2 is a transposition (12) is again obvious.

So assume S2 is a product of two disjoint transpositions. Thus S2 is an element of the alternating group on four letters. But if Γ is 3[8]2 then S1, an element of order three, is also in the alternating group and hence so is G=S1,S2; a contradiction. If Γ is 4[6]2 then as mentioned before S1S2 has order three and hence is in the alternating group. This again is impossible since S1 has order four and does not lie in the alternating group.

Using Corollary 9 we can now proceed to verify our conjecture (11).

If wW, w1, and θ(w) is not a reflection, then w is not conjugate to any element of either r1 or r2 and μ(w)=0 so k(w)=2. Hence Ψ(w)=1= (-1)k(w) m1μ(w).

If θ(w) is a reflection, then by Theorem 5, w is conjugate to an element of r1 or an element of r2. By looking at the matrices one easily sees that if xri, x1, then w(x)=w(ri) and thus by Corollary 9, w(x)=ri×𝒵(W). Further from [Cox1974] we know that |𝒵(W)|(q,2)hq and that m1=2hq-1. We next separate two cases.

Case (i) q is even. Without loss of generality we assume w is to an element of r1. By Corollary 6 w is not conjugate to an element of r2. Hence 1r2W(w)=0. Since S1, S12, , S1p1-1 have distinct non identity eigenvalues the conjugacy classes determined by r1, r12, , r1p1-1 are all distinct. Hence 1r1W (w) = 1p1|W(w)| = |𝒵(W)| = 2hq So Ψ(w)=1-2hq=-m1=(-1)k(w)m1μ(w).

Case (ii) q is odd. Thus, p1=p2=p, say and r1 is conjugate to r2; so w is conjugate to an element of r1 and to an element of r2. Hence 1r1W (w)=1r2W (w)=1p |W(w)|= hq, So again, Ψ(w)=1-2hq=-m1=(-1)k(w)m1μ(w). This completes the verification of our conjecture in case =2.

Let Γ𝒞+. Put W=W(Γ) and let I denote the vertex set of Γ. Let Bi={wW|μ(w)=-1} 0i and put bi=|Bi|. As usual we let mi=di-1 where d1-1d-1 are the exponents of θ(W). In [STo1954] Shephard and Todd introduced and verified and later in [Sol1963] Solomon gave a proof for the polynomial identity: (13) i=0bi ti=i=1 (1+mit).

For JI let V(J)V be the subspace of V spanned by the basis vectors corresponding to the vertices in J. Also let m1Jm2Jm|J|J be the exponents of θ(W(J)) acting on V(J). Further defining μJ on W(J) as we defined μ on W and putting kJ=|J|-μJ one easily sees that if xW(J) then k(x)=kJ(x). Finally define a class function τ:W[t] by τ(w)=tk(w). Notice that the left hand side of (13) can be written as wWτ(w).

Consider the sum wW Ψ(w)τ(w) = wWτ(w) JI(-1)J 1W(J)W(w) = JI(-1)J wW1W(J)W (w)τ(w) = JI(-1)J |W|[1W(J)W,τ] = JI(-1)J |W|[1W(J),τW(J)] = JI(-1)J |W:W(J)| wW(J) τ(w) = JI(-1)J |W:W(J)| i=1|J| (1+mijt) In the last step we have used (13) applied to the groups W(J).

Next consider the sum wW (-1)k(w) m1μ(w) tk(2) = wW (-1)k(w) m1-k(w) tk(w) = m1wW (-tm1)k(w) = m1i=1 (1-mim1t) = i=1 (m1-mit) Here, in the next to the last step we have used (13).

So, if our conjecture (11) is true, we have a new polynomial identity: (14) JI(-1)J |W:W(J)| i=1|J| (1+mijt)= i=1 (m1-mit). In case W is a Coxeter group, m1=1 and (14) becomes Corollary 2.4 of [Sol1966]. We have verified (14) in a case by case manner for all the non Coxeter groups associated with connected graphs in 𝒞+ with the exception of the infinite family Bp. This lends another measure of credibility to our conjecture (11).

Notes and references

This is a typed version of David W. Koster's thesis Complex Reflection Groups.

This thesis was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) at the University of Wisconsin - Madison, 1975.

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