Complex Reflection Groups

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

Last update: 4 January 2014


The classification of the finite real reflection groups involves the study of group presentations in which there are a finite number of generators r1,,r satisfying relations of the form (rirj)mij=1 with mii=1 for 1i. A comprehensive treatement of such groups, known as Coxeter groups after H.S.M. Coxeter, occurs in [Bou1968]. In 1953 Shephard and Todd classified all finite complex reflection groups [STo1954]. Then in 1966 Coxeter gave presentations for all the dimensional finite complex reflection groups generated by reflections [Cox1967]. Most of these presentations can be described as follows. There are a finite number of generators, r1,,r, and a symmetric × matrix, M=(qij), in which each entry is a positive integer qij2. We denote the diagonal entries qii of M by pi. This matrix defines the relations in our group presentation. For each diagonal entry pi there is a relation ripi=1 and for each off diagonal entry qij there is a relation rirjri=rjrirj with qij factors
on each side of the equal sign. Notice that if pi=pj=2, then the relation rirjri=rjrirj with qij symbols on each side of the
equal sign is equivalent to the relation (rirj)qij=1. Thus we see that the group presentations encountered in the study of real reflection groups are present among those just described.

In [Cox1967] Coxeter introduced a graphical notation for this type of group presentation. If the presentation has generators r1,,r, so the matrix M=(qij) is ×, the graph has vertices. Corresponding to the diagonal entry pi of M the number pi is written under the ith vertex except when pi=2 and then the vertex remains unlabeled. For each off diagonal entry qij that is greater than 2 an edge is drawn between the ith and jth vertices and this edge is labeled with the number qij except when qij=3 and then the edge remains unlabeled. If qij=2 so ri and rj commute, no edge is drawn between the ith and jth vertices. For example, the graph p 4 indicates the group presentation r1,r2,r3| r1p=r22=r32 =1,r1r2r1r2 =r2r1r2r1, r1r3=r3r1, r2r3r2=r3r2 r3 which is given by the matrix M=(p42423232). Using the symbol Γ to denote such a graph we let W(Γ) denote the corresponding abstract group.

We remark here that if qij is odd, then the relation rirjri=rjrirj can be viewed as (rirj)qij-12 ri=rj (rirj)qij-12 forcing ri to be conjugate to rj and hence these two generators must have the same order. Looking ahead to Theorem 1 and its Corollary we see that in this case we must require pi=pj for without this restriction Theorem 1 is false. We thus denote by the collection of all symmetric matrices M=(qij) such that each entry is a positive integer qij2 and if an off diagonal entry qij is odd then the diagonal entries pi=qii and pj=qjj are the same. For M we let Γ(M) denote the graph of the group presentation given by M and we let 𝒞={Γ(M)|M} be the collection of all those graphs associated with the matrices in . Finally we denote by 𝒞 the subclass of all those graphs in 𝒞 with vertices. We will be interested in studying the groups W(Γ) for Γ𝒞.

The first result obtained for the groups W(Γ) is stated precisely in Theorem 1 and can be summarized as follows: Let Γ𝒞 and let V be an dimensional complex vector space. Then there is a homomorphism θ:W(Γ)GL(V) and a Hermitian form H(Γ) on V which is invariant under the linear group θ(W(Γ)). Further θ(ri) is a reflection of order pi for all 1i.

We begin Section II by showing that for Γ𝒞 the group W(Γ) admits an antiautomorphism, denoted ww, which fixes the generators r1,,r. Thus defining ρ:W(Γ)GL(V) by ρ(w)=θ(w)t we see that ρ is another representation of W(Γ). Now if Γ is a linear graph (see definition on page 15) Coxeter gives a representation of W(Γ) [Cox1967] and we point out that Coxeter's representation is the ρ defined above. If Γ𝒞 is such that pi=2 for all 1i (so W(Γ) is a Coxeter group) then the form H(Γ) and the representation θ of Theorem 1 are precisely the form "B" and representation "σ" given in [Bou1968].

For Γ𝒞 a connected graph we show in Proposition 6 that a necessary condition for the finiteness of W(Γ) is that H(Γ) be positive definite and a large portion of Section II is devoted to the classification of all those connected graphs Γ𝒞 with H(Γ) positive definite. This result given in Theorem 2. Then a comparison with those graphs listed by Coxeter [Cox1967] yields that the positive definiteness of H(Γ) is also a sufficient condition for W(Γ) to be finite. This is stated in Theorem 4. For the final result of Section II we use the relationship between the representation θ of Theorem 1 and the representations σ[1] and ρ[3] to obtain the result that if W(Γ) is finite, then θ is a faithful representation. We remark that this is false if the hypothesis of the finiteness of W(Γ) is deleted.

In Section III we use a result of Springer [Spr1974] to show that if Γ𝒞 and W(Γ) is finite then every reflection in θ(W(Γ)) is conjugate in θ(W(Γ)) to a power of one of the generating reflections θ(ri), 1i. For Coxeter groups W(Γ) this is a known result [Bou1968].

In Section IV we begin by verifying that for a connected graph Γ𝒞 with W(Γ) finite one can obtain a certain set of integers, called the exponents of θ(W(Γ)) (see page 47 or [STo1954]) from knowledge of the eigenvalues of the inverse of the group element S=θ(r1)··θ(r). Then letting h denote the order of the element S we show that the number of reflections in θ(W(Γ)) is hk=1 pk-1pk This result in combination with an observation of McMullen produces the fact that the number of reflections in θ(W(Γ)) plus the number of reflecting hyperplanes in V is equal to h. These results are stated precisely in Theorems 6 and 7 and in Corollaries 7 and 8.

For connected Γ𝒞 with W(Γ) finite we let d-1 denote the smallest exponent of θ(W(Γ)) and we let I be the set of vertices of Γ. For JI we let Γ(J) be the subgraph of Γ obtained by deleting from Γ all those vertices in I\J and all the edges connected to those vertices. We put W(J)=W(Γ(J)). We end section IV by verifying that JI (-1)|J| |W(Γ):W(J)| =(d-1). We generalize this arithmetic formula to a conjecture about characters induced from the principal characters 1W(J) and we give some evidence supporting the conjecture.

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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