Complex Reflection Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 4 January 2014

Introduction

The classification of the finite real reflection groups involves the study of group presentations in which there are a finite number of generators $r_1,\dots ,r_{\ell }$ satisfying relations of the form ${\left(r_i{r}_j\right)}^{m_{ij}}=1$ with $m_{ii}=1$ for $1\le i\le \ell \text{.}$ 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 $\ell$ dimensional finite complex reflection groups generated by $\ell$ reflections [Cox1967]. Most of these presentations can be described as follows. There are a finite number of generators, $r_1,\dots ,r_{\ell },$ and a symmetric $\ell \times \ell$ matrix, $M=\left(q_{ij}\right),$ in which each entry is a positive integer $q_{ij}\ge 2\text{.}$ We denote the diagonal entries $q_{ii}$ of $M$ by $p_i\text{.}$ This matrix defines the relations in our group presentation. For each diagonal entry $p_i$ there is a relation $r_i^{p_i}=1$ and for each off diagonal entry $q_{ij}$ there is a relation $r_i{r}_j{r}_i\cdots =r_j{r}_i{r}_j\cdots$ with $q_{ij}$ factors
on each side of the equal sign. Notice that if $p_i=p_j=2,$ then the relation $r_i{r}_j{r}_i\cdots =r_j{r}_i{r}_j\cdots$ with $q_{ij}$ symbols on each side of the
equal sign is equivalent to the relation ${\left(r_i{r}_j\right)}^{q_{ij}}=1\text{.}$ 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 $\ell$ generators $r_1,\dots ,r_{\ell },$ so the matrix $M=\left(q_{ij}\right)$ is $\ell \times \ell ,$ the graph has $\ell$ vertices. Corresponding to the diagonal entry $p_i$ of $M$ the number $p_i$ is written under the $i^{\text{th}}$ vertex except when $p_i=2$ and then the vertex remains unlabeled. For each off diagonal entry $q_{ij}$ that is greater than $2$ an edge is drawn between the $i^{\text{th}}$ and $j^{\text{th}}$ vertices and this edge is labeled with the number $q_{ij}$ except when $q_{ij}=3$ and then the edge remains unlabeled. If $q_{ij}=2$ so $r_i$ and $r_j$ commute, no edge is drawn between the $i^{\text{th}}$ and $j^{\text{th}}$ vertices. For example, the graph $$\begin{array}{c} p 4 \end{array}$$ indicates the group presentation $$⟨r_1,r_2,r_3\hspace{0.17em}|\hspace{0.17em}{r}_1^p=r_2^2=r_3^2=1,\hspace{0.17em}{r}_1{r}_2{r}_1{r}_2=r_2{r}_1{r}_2{r}_1,\hspace{0.17em}{r}_1{r}_3=r_3{r}_1,\hspace{0.17em}{r}_2{r}_3{r}_2=r_3{r}_2{r}_3⟩$$ which is given by the matrix $M=\left(\begin{array}{ccc}p& 4& 2\\ 4& 2& 3\\ 2& 3& 2\end{array}\right)\text{.}$ Using the symbol $\Gamma$ to denote such a graph we let $W\left(\Gamma \right)$ denote the corresponding abstract group.

We remark here that if $q_{ij}$ is odd, then the relation $r_i{r}_j{r}_i\cdots =r_j{r}_i{r}_j\cdots$ can be viewed as $${\left(r_i{r}_j\right)}^{\frac{q_{ij}-1}{2}}{r}_i=r_j{\left(r_i{r}_j\right)}^{\frac{q_{ij}-1}{2}}$$ forcing $r_i$ to be conjugate to $r_j$ 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 $p_i=p_j$ for without this restriction Theorem 1 is false. We thus denote by $ℳ$ the collection of all symmetric matrices $M=\left(q_{ij}\right)$ such that each entry is a positive integer $q_{ij}\ge 2$ and if an off diagonal entry $q_{ij}$ is odd then the diagonal entries $p_i=q_{ii}$ and $p_j=q_{jj}$ are the same. For $M\in ℳ$ we let $\Gamma \left(M\right)$ denote the graph of the group presentation given by $M$ and we let $𝒞=\left\{\Gamma \left(M\right)\hspace{0.17em}\right|\hspace{0.17em}M\in ℳ\}$ be the collection of all those graphs associated with the matrices in $ℳ\text{.}$ Finally we denote by ${𝒞}_{\ell }$ the subclass of all those graphs in $𝒞$ with $\ell$ vertices. We will be interested in studying the groups $W\left(\Gamma \right)$ for $\Gamma \in 𝒞\text{.}$

The first result obtained for the groups $W\left(\Gamma \right)$ is stated precisely in Theorem 1 and can be summarized as follows: Let $\Gamma \in {𝒞}_{\ell }$ and let $V$ be an $\ell$ dimensional complex vector space. Then there is a homomorphism $\theta :W\left(\Gamma \right)\to GL\left(V\right)$ and a Hermitian form $H\left(\Gamma \right)$ on $V$ which is invariant under the linear group $\theta \left(W\left(\Gamma \right)\right)\text{.}$ Further $\theta \left(r_i\right)$ is a reflection of order $p_i$ for all $1\le i\le \ell \text{.}$

We begin Section II by showing that for $\Gamma \in 𝒞$ the group $W\left(\Gamma \right)$ admits an antiautomorphism, denoted $w\to w\prime ,$ which fixes the generators $r_1,\dots ,r_{\ell }\text{.}$ Thus defining $\rho :W\left(\Gamma \right)\to GL\left(V\right)$ by $\rho \left(w\right)=\theta {\left(w\prime \right)}^t$ we see that $\rho$ is another representation of $W\left(\Gamma \right)\text{.}$ Now if $\Gamma$ is a linear graph (see definition on page 15) Coxeter gives a representation of $W\left(\Gamma \right)$ [Cox1967] and we point out that Coxeter's representation is the $\rho$ defined above. If $\Gamma \in {𝒞}_{\ell }$ is such that $p_i=2$ for all $1\le i\le \ell$ (so $W\left(\Gamma \right)$ is a Coxeter group) then the form $H\left(\Gamma \right)$ and the representation $\theta$ of Theorem 1 are precisely the form $\text{"}B\text{"}$ and representation $\text{"}\sigma \text{"}$ given in [Bou1968].

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

In Section III we use a result of Springer [Spr1974] to show that if $\Gamma \in {𝒞}_{\ell }$ and $W\left(\Gamma \right)$ is finite then every reflection in $\theta \left(W\left(\Gamma \right)\right)$ is conjugate in $\theta \left(W\left(\Gamma \right)\right)$ to a power of one of the generating reflections $\theta \left(r_i\right),$ $1\le i\le \ell \text{.}$ For Coxeter groups $W\left(\Gamma \right)$ this is a known result [Bou1968].

In Section IV we begin by verifying that for a connected graph $\Gamma \in {𝒞}_{\ell }$ with $W\left(\Gamma \right)$ finite one can obtain a certain set of integers, called the exponents of $\theta \left(W\left(\Gamma \right)\right)$ (see page 47 or [STo1954]) from knowledge of the eigenvalues of the inverse of the group element $S=\theta \left(r_1\right)·\dots ·\theta \left(r_{\ell }\right)\text{.}$ Then letting $h$ denote the order of the element $S$ we show that the number of reflections in $\theta \left(W\left(\Gamma \right)\right)$ is $$h\sum _{k=1}^{\ell }\frac{p_k-1}{p_k}$$ This result in combination with an observation of McMullen produces the fact that the number of reflections in $\theta \left(W\left(\Gamma \right)\right)$ plus the number of reflecting hyperplanes in $V$ is equal to $\ell h\text{.}$ These results are stated precisely in Theorems 6 and 7 and in Corollaries 7 and 8.

For connected $\Gamma \in {𝒞}_{\ell }$ with $W\left(\Gamma \right)$ finite we let $d-1$ denote the smallest exponent of $\theta \left(W\left(\Gamma \right)\right)$ and we let $I$ be the set of vertices of $\Gamma \text{.}$ For $J\subset I$ we let $\Gamma \left(J\right)$ be the subgraph of $\Gamma$ obtained by deleting from $\Gamma$ all those vertices in $I\backslash J$ and all the edges connected to those vertices. We put $W\left(J\right)=W\left(\Gamma \left(J\right)\right)\text{.}$ We end section IV by verifying that $$\sum _{J\subset I}{(-1)}^{\left|J\right|}|W\left(\Gamma \right):W\left(J\right)|={(d-1)}^{\ell }\text{.}$$ We generalize this arithmetic formula to a conjecture about characters induced from the principal characters $1_{W\left(J\right)}$ 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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