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 ${r}_{1},\dots ,{r}_{\ell}$ 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 $$\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\np\n4\n\n\end{array}$$ indicates the group presentation $$\u27e8{r}_{1},{r}_{2},{r}_{3}\hspace{0.17em}|\hspace{0.17em}{r}_{1}^{p}={r}_{2}^{2}={r}_{3}^{2}=1,\hspace{0.17em}{r}_{1}{r}_{2}{r}_{1}{r}_{2}={r}_{2}{r}_{1}{r}_{2}{r}_{1},\hspace{0.17em}{r}_{1}{r}_{3}={r}_{3}{r}_{1},\hspace{0.17em}{r}_{2}{r}_{3}{r}_{2}={r}_{3}{r}_{2}{r}_{3}\u27e9$$ 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 $\mathcal{M}$ 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 $M\in \mathcal{M}$ we let $\Gamma \left(M\right)$ denote the graph of the group presentation given by $M$ and we let $\mathcal{C}=\left\{\Gamma \left(M\right)\hspace{0.17em}\right|\hspace{0.17em}M\in \mathcal{M}\}$ be the collection of all those graphs associated with the matrices in $\mathcal{M}\text{.}$ Finally we denote by ${\mathcal{C}}_{\ell}$ the subclass of all those graphs in $\mathcal{C}$ with $\ell $ vertices. We will be interested in studying the groups $W\left(\Gamma \right)$ for $\Gamma \in \mathcal{C}\text{.}$

The first result obtained for the groups $W\left(\Gamma \right)$ is stated precisely in Theorem 1 and can be summarized as follows: Let $\Gamma \in {\mathcal{C}}_{\ell}$ 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 $\Gamma \in \mathcal{C}$ 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 $\Gamma \in {\mathcal{C}}_{\ell}$ 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 $\Gamma \in \mathcal{C}$ 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 $\Gamma \in \mathcal{C}$ 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 $\Gamma \in {\mathcal{C}}_{\ell}$ 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 $\Gamma \in {\mathcal{C}}_{\ell}$ 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 $S=\theta \left({r}_{1}\right)\xb7\dots \xb7\theta \left({r}_{\ell}\right)\text{.}$ 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 $\Gamma \in {\mathcal{C}}_{\ell}$ 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.

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.