Last update: 10 January 2014

Let $V$ be an $\ell $ dimensional complex vector space and let $P$ denote the ring of polynomial functions on $V\text{.}$ If $G\subseteq GL\left(V\right),$ then $G$ acts naturally on $P$ and the elements of $P$ invariant under $G$ form a subring $I\left(P\right)$ of $P\text{.}$ In [STo1954] it is shown that $G$ is a finite group generated by reflections if and only if $I\left(P\right)$ is generated by $\ell $ algebraically independent homogeneous polynomials ${f}_{1},\dots ,{f}_{\ell}\text{.}$ Let ${d}_{i}$ denote the degree of ${f}_{i}\text{.}$ The integers ${d}_{1},\dots ,{d}_{\ell}$ are uniquely determined by $G$ and the numbers ${d}_{1}-1,\dots ,{d}_{\ell}-1$ are called the exponents of group $G\text{.}$ From [STo1954] we have $\left|G\right|=\prod _{i=1}^{\ell}{d}_{i}$ and that the number of reflections in $G$ is $\sum _{i=1}^{\ell}({d}_{i}-1)\text{.}$ In [STo1954], Shephard and Todd further show that if $G$ may be generated by $\ell $ reflections then one may choose generating reflections ${R}_{1},\dots ,{R}_{\ell}$ so that the eigenvalues of the product ${R}_{1},\dots ,{R}_{\ell}$ are ${\gamma}^{{d}_{1}-1},\dots ,{\gamma}^{{d}_{\ell}-1}$ where $\gamma ={e}^{2\pi i/h}$ and ${d}_{\ell}=h$ is the order of ${R}_{1}\dots {R}_{\ell}\text{.}$ In their argument there is no general algorithm for choosing such generators. However, if $G=\theta \left(W\left(\Gamma \right)\right),$ for some connected $\Gamma \in {\mathcal{C}}_{\ell}^{+}$ then there is a natural choice for such generators.

**Theorem 6.** Let $\Gamma \in {\mathcal{C}}_{\ell}^{+}$
be connected. As usual we let ${r}_{1},\dots ,{r}_{\ell}$
be the generators for $W\left(\Gamma \right)$ and we let
${S}_{i}=\theta \left({r}_{i}\right)\text{.}$
Put ${R}_{i}={S}_{i}^{-1}$ and let
$\Pi $ be any permutation of $\{1,\dots ,\ell \}\text{.}$
Then

(i) | The conjugacy class of ${r}_{\Pi \left(1\right)}\xb7\dots \xb7{r}_{\Pi \left(\ell \right)}$ does not depend on $\Pi \text{.}$ |

(ii) | The eigenvalues of ${R}_{1}\xb7\dots \xb7{R}_{\ell}$ are ${\gamma}^{{d}_{1}-1},\dots ,{\gamma}^{{d}_{\ell}-1}$ where $\gamma ={e}^{2\pi i/h}$ and $h$ is the order of ${R}_{1}\xb7\dots \xb7{R}_{\ell}\text{.}$ |

(iii) | ${d}_{\ell}=h\text{.}$ |

Proof. | |

By Theorem 2 $\Gamma $ is a tree and thus (i) is given in [Bou1968] (Lemma 1, Chapter V, section 6). For a graph $\Gamma $ with just one vertex labelled with the integer $p$ we see from [STo1954] that ${d}_{1}=p$ and thus (ii) and (iii) are obvious. Now if $\Gamma $ satisfies ${p}_{i}=2,$ $1\le i\le \ell ,$ so $W\left(\Gamma \right)$ is a Coxeter group, (ii) and (iii) are given by a general argument in [Bou1968] (Proposition 3, Chapter V, section 6). If $\Gamma \in {\mathcal{C}}_{2}^{+},$ say $\Gamma $ is ${p}_{1}\left[q\right]{p}_{2},$ Coxeter observes [Cox1962] that ${d}_{1}=2h/q$ and ${d}_{2}=h\text{.}$ He also obtains $\gamma $ and ${\gamma}^{1+h-\frac{2h}{q}}$ as the eigenvalues of ${S}_{1}^{t}{S}_{2}^{t}$ and thus we have ${\gamma}^{\frac{2h}{q}-1}$ and ${\gamma}^{h-1}$ as the eigenvalues of ${R}_{1}{R}_{2}\text{.}$ 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 $\Gamma $ is $3\left[3\right]3\left[3\right]3\text{.}$ By comparing $\left[3\right]$ with $\left[2\right]$ we see that ${d}_{1}=6,$ ${d}_{2}=9,$ ${d}_{3}=12\text{.}$ Now the matrix for ${R}_{1}{R}_{2}{R}_{3}$ in the basis $\{{v}_{1},{v}_{2},{v}_{3}\}$ is $\left(\begin{array}{ccc}0& 0& -\omega \\ \alpha & 0& {\omega}^{2}\alpha \\ 0& \alpha & {\omega}^{2}\end{array}\right)$ where $\omega ={e}^{2\pi i/3}$ and $\alpha =\frac{1-{\omega}^{2}}{\sqrt{3}}\text{.}$ The characteristic equation is thus ${\lambda}^{3}-{\omega}^{2}{\lambda}^{2}+\omega \lambda -1=0$ which has roots ${\gamma}^{5},$ ${\gamma}^{8},$ ${\gamma}^{11}$ where $\gamma ={e}^{2\pi i/12}\text{.}$ For the graph $\Gamma $ $3\left[3\right]3\left[4\right]2$ we find from $\left[3\right]$ and $\left[2\right]$ that the degrees are ${d}_{1}=6,$ ${d}_{2}=12,$ ${d}_{3}=18\text{.}$ We compute that the matrix for ${R}_{1}{R}_{2}{R}_{3}$ in the basis $\{{v}_{1},{v}_{2},{v}_{3}\}$ is $\left(\begin{array}{ccc}0& \alpha & B{\omega}^{2}\\ \alpha & 1& -\alpha B\\ 0& B& -1\end{array}\right)$ where $\alpha =\frac{1-{\omega}^{2}}{\sqrt{3}}$ and $B={3}^{\frac{1}{4}}\text{.}$ Thus the characteristic equation is ${\lambda}^{3}+\omega =0$ and the eigenvalues of ${R}_{1}{R}_{2}{R}_{3}$ are ${\gamma}^{5},$ ${\gamma}^{11},$ ${\gamma}^{17}$ where $\gamma ={e}^{2\pi i/18}\text{.}$ If $\Gamma $ is $3\left[3\right]3\left[3\right]3\left[3\right]3$ we find again from $\left[3\right]$ and $\left[2\right]$ that ${d}_{1}=12,$ ${d}_{2}=18,$ ${d}_{3}=24,$ ${d}_{4}=30\text{.}$ This time the matrix for ${R}_{1}{R}_{2}{R}_{3}{R}_{4}$ in the basis $\{{v}_{1},{v}_{2},{v}_{3},{v}_{4}\}$ is $\left(\begin{array}{cccc}0& 0& 0& -\omega \alpha \\ \alpha & 0& 0& -\omega \\ 0& \alpha & 0& {\omega}^{2}\alpha \\ 0& 0& \alpha & {\omega}^{2}\end{array}\right)$ where $\omega ={e}^{2\pi i/3}$ and $\alpha =\frac{1-{\omega}^{2}}{\sqrt{3}}\text{.}$ The characteristic equation is thus ${\lambda}^{4}-{\omega}^{2}{\lambda}^{3}+\omega {\lambda}^{2}-\lambda +{\omega}^{2}=0$ which has roots ${\gamma}^{11},$ ${\gamma}^{17},$ ${\gamma}^{23},$ ${\gamma}^{29}$ where $\gamma ={e}^{2\pi i/30}\text{.}$ We finally consider the infinite family of graphs ${B}_{\ell}^{p}\text{.}$ Here, comparing $\left[3\right]$ and $\left[2\right]$ yields that the degrees are given by ${d}_{k}=kp$ $(1\le k\le \ell )\text{.}$ Letting $\alpha ={\{2\text{sin}\hspace{0.17em}\pi /p\}}^{-\frac{1}{2}}$ we define a new basis $\{{x}_{1},\dots ,{x}_{\ell}\}$ by ${x}_{i}=\alpha {v}_{1}+{v}_{2}+{v}_{3}+\dots +{v}_{i}\text{.}$ Then $$\begin{array}{c}{S}_{1}\left({x}_{1}\right)=\epsilon {x}_{1}\phantom{\rule{2em}{0ex}}(\epsilon ={e}^{2\pi i/p})\\ {S}_{1}\left({x}_{j}\right)={x}_{j}\phantom{\rule{2em}{0ex}}\text{for}\hspace{0.17em}j\ne i\text{.}\end{array}$$ Further if $2\le k\le \ell ,$ ${S}_{k}$ interchanges ${x}_{k-1}$ and ${x}_{k}$ and fixes every other ${x}_{j}\text{.}$ Hence the matrix for ${R}_{1}{R}_{2}\xb7\dots \xb7{R}_{\ell}\text{.}$ in this basis is $$\left(\begin{array}{ccccccc}0& & & & & & {\epsilon}^{-1}\\ 1& 0& & & & \\ & 1& 0\\ & & 1\\ & & & & & 0\\ & & & & & 1& 0\end{array}\right)\text{.}$$ The characteristic equation is thus ${\lambda}^{\ell}-{\epsilon}^{-1}=0\text{.}$ Hence $h=\ell p$ and letting $\gamma ={e}^{2\pi i/h}$ the roots are ${\gamma}^{kp-1}$ $1\le k\le \ell \text{.}$ This completes the verification. $\square $ |

Let $\Gamma \in {\mathcal{C}}_{\ell}^{+}$ be connected. Put $S={S}_{1}\xb7\dots \xb7{S}_{\ell},$ let $h$ denote the order of $S$ and let $\gamma ={e}^{2\pi i/h}\text{.}$ We define the integers ${n}_{1},\dots ,{n}_{\ell}$ $(0\le {n}_{1}\le \dots \le {n}_{\ell}\le h-1)$ and ${m}_{1},\dots ,{m}_{\ell}$ $(0\le {m}_{1}\le {m}_{2}\le \dots \le {m}_{\ell}\le h-1)$ by requiring that ${\gamma}^{{n}_{j}},$ $1\le j\le \ell $ are the eigenvalues for $S$ and ${\gamma}^{{m}_{j}},$ $1\le j\le \ell ,$ are the eigenvalues for ${S}^{-1}\text{.}$ Thus, using Theorem 6, ${m}_{i}={d}_{i}-1\text{.}$ In order to give a general argument for Theorem 6 along the same lines as that given in [Bou1968] for the case where $W\left(\Gamma \right)$ is a Coxeter group one needs to know three things:

(a) | ${n}_{1}=1$ |

(b) | There is an eigenvector corresponding to the eigenvalue $\gamma $ of $S$ which does not lie in the reflecting hyperplane of any reflection in $\theta \left(W\left(\Gamma \right)\right)\text{.}$ |

(c) | The number of reflections in $\theta \left(W\left(\Gamma \right)\right)$ is $\sum _{i=1}^{\ell}{m}_{i}\text{.}$ |

The quantity $\sum _{k=1}^{\ell}{\omega}_{k}$ arises also in another setting. We have $\text{Det}\left({S}^{-1}\right)={e}^{\frac{2\pi i}{h}\sum _{k=1}^{\ell}{m}_{k}}\text{.}$ On the other hand, $$\text{Det}\left({S}^{-1}\right)=\prod _{k=1}^{\ell}\text{Det}\left({S}_{k}^{-1}\right)={e}^{2\pi i\sum _{k=1}^{\ell}\frac{{p}_{k}-1}{{p}_{k}}}\text{.}$$ Thus, $$\begin{array}{cc}\text{(7)}& \frac{1}{h}\sum _{k=1}^{\ell}{m}_{k}\equiv \sum _{k=1}^{\ell}\frac{{p}_{k}-1}{{p}_{k}}\phantom{\rule{1em}{0ex}}\text{mod}\hspace{0.17em}\mathbb{Z}\text{.}\end{array}$$ Similarly, by computing $\text{Det}\left(S\right)$ in two ways we obtain $$\begin{array}{cc}\text{(8)}& \frac{1}{h}\sum _{k=1}^{\ell}{n}_{k}\equiv \sum _{k=1}^{\ell}\frac{1}{{p}_{k}}\phantom{\rule{1em}{0ex}}\text{mod}\hspace{0.17em}\mathbb{Z}\text{.}\end{array}$$ In case $W\left(\Gamma \right)$ is a Coxeter group, all ${p}_{i}=2,$ $S$ is conjugate to ${S}^{-1},$ and both (7) and (8) become $$\begin{array}{cc}\text{(9)}& \sum _{k=1}^{\ell}{m}_{k}\equiv \sum _{k=1}^{\ell}{n}_{k}\equiv \frac{\ell h}{2}\phantom{\rule{1em}{0ex}}\text{mod}\hspace{0.17em}\mathbb{Z}\text{.}\end{array}$$ These congruences are in fact equalities [Bou1968, p.118] and this leads one to suspect that (7) and (8) are also equalities. Since ${m}_{k}+{n}_{\ell -k+1}=h$ equality holds in (7) if and only if it holds in (8).

**Theorem 7.** Let $\Gamma \in {\mathcal{C}}_{\ell}^{+}$ be
connected. With ${m}_{1},\dots ,{m}_{\ell}$ and
$h$ defined as above we have
$$\frac{1}{h}\sum _{k=1}^{\ell}{m}_{k}=\sum _{k=1}^{\ell}\frac{{p}_{k}-1}{{p}_{k}}\text{.}$$

Proof. | |

We may assume the ${p}_{k}$ are not all $2\text{.}$ If $\ell =1$ the statement is obvious. If $\ell =2,$ $\Gamma $ is ${p}_{1}\left[q\right]{p}_{2}\text{.}$ From [Cox1962] we have that ${m}_{1}=\frac{2h}{q}-1,$ ${m}_{2}=h-1,$ and $h=\frac{2{p}_{1}{p}_{2}q}{({p}_{1}+{p}_{2})q-{p}_{1}{p}_{2}(q-2)}\text{.}$ Thus we have $$\begin{array}{ccc}\frac{1}{h}({m}_{1}+{m}_{2})& =& \frac{2}{q}-\frac{1}{h}+1-\frac{1}{h}=\frac{2}{q}+1-\frac{2}{h}\\ & =& \frac{2}{q}+1-\frac{{p}_{1}+{p}_{2}}{{p}_{1}{p}_{2}}+1-\frac{2}{q}\\ & =& (1-\frac{1}{{p}_{1}})+1-\frac{1}{{p}_{2}}\\ & =& \frac{{p}_{1}-1}{{p}_{1}}+\frac{{p}_{2}-1}{{p}_{2}}\text{.}\end{array}$$ If $\ell \ge 3$ we have just computed the numbers ${m}_{1},\dots ,{m}_{\ell}$ and $h$ in the verification of Theorem 6. It thus becomes a trivial arithmetic task to verify the desired result for the remaining cases. $\square $ |

**Corollary 7.** Let $\Gamma \in {\mathcal{C}}_{\ell}^{+}$ be
connected. With $h$ defined as above we have that the number of reflections in $\theta \left(W\left(\Gamma \right)\right)$
is $$h\sum _{k=1}^{\ell}\frac{{p}_{k}-1}{{p}_{k}}\text{.}$$

Proof. | |

Since ${m}_{i}={d}_{i}-1,$ and $\sum _{i=1}^{\ell}({d}_{i}-1)$ is the number of reflections in $\theta \left(W\left(\Gamma \right)\right)$ this follows immediately from Theorem 7. $\square $ |

Coxeter [Cox1974, p.153] credits McMullen with the observation that for a connected graph $\Gamma \in {\mathcal{C}}_{\ell}^{+}$ the number of reflecting hyperplanes in $V$ corresponding to reflections in $\theta \left(W\left(\Gamma \right)\right)$ is $h\sum _{k=1}^{\ell}\frac{1}{{p}_{k}}\text{.}$ 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 $\Gamma \in {\mathcal{C}}_{\ell}^{+}$
be connected. If $h$ is defined as above then the number of reflecting hyperplanes in $V$ plus the number of reflections in
$\theta \left(W\left(\Gamma \right)\right)$ is
$\ell h\text{.}$

Let $\Gamma \in {\mathcal{C}}_{\ell}$ and let $I$ denote the set of vertices of $\Gamma \text{.}$ For $J\subset I$ we denote by $\Gamma \left(J\right)$ the subgraph of $\Gamma $ obtained by deleting from $\Gamma $ all those vertices in $I\backslash J$ and the edges connected to those vertices. Also we put $W\left(J\right)=W\left(\Gamma \left(J\right)\right)\text{.}$ Here we agree that if $J=\varphi ,$ then $W\left(J\right)=1\text{.}$ Finally we write ${(-1)}^{J}$ for ${(-1)}^{\left|J\right|}\text{.}$

**Proposition 10.** Let $\Gamma \in {\mathcal{C}}_{\ell}^{+}$
be connected. Put $W=W\left(\Gamma \right)$ and let
$I$ denote the vertex set of $\Gamma \text{.}$ Put
${m}_{i}={d}_{i}-1$ where
${d}_{1}-1\le \dots \le {d}_{\ell}-1$
are the exponents of $\theta \left(W\right)\text{.}$ Then
$$\begin{array}{cc}\text{(10)}& \sum _{J\subset I}{(-1)}^{J}|W:W\left(J\right)|={m}_{1}^{\ell}\text{.}\end{array}$$

Proof. | |

If $W$ is a Coxeter group, ${m}_{1}=1$ and thus (10) occurs in [Sol1966, p.378] where Solomon credits it to Witt. If $\Gamma $ has just one vertex labelled with the integer $p,$ we have $\left|W\right|=p$ and ${m}_{1}=p-1$ so (10) is trivial. Suppose next that $\Gamma $ has two vertices; say $\Gamma $ is ${p}_{1}\left[q\right]{p}_{2}\text{.}$ Letting $h$ denote the order of ${S}_{1}{S}_{2}$ we know from Theorem 6 that $h={m}_{2}+1$ and thus from Theorem 7 that $1-\frac{1}{{p}_{1}}-\frac{1}{{p}_{2}}=\frac{{m}_{1}-1}{{m}_{2}+1}\text{.}$ Using this together with the previously mentioned fact that $\left|W\right|=({m}_{1}+1)({m}_{2}+1)$ we compute $$\begin{array}{ccc}\sum _{J\subset I}{(-1)}^{J}|W:W\left(J\right)|& =& \left|W\right|(1-\frac{1}{{p}_{1}}-\frac{1}{{p}_{2}})+1\\ & =& ({m}_{1}+1)({m}_{2}+1)\frac{({m}_{1}-1)}{{m}_{2}+1}+1\\ & =& {m}_{1}^{2}\text{.}\end{array}$$ Now ignoring the infinite family ${B}_{\ell}^{p}$ 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 $\Gamma $ is the graph $${B}_{\ell}^{p}\phantom{\rule{2em}{0ex}}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\np\n4\n\cdots \n\n\end{array}$$ Consulting [STo1954] and [Cox1967] we see that $\left|W\right|={p}^{\ell}\ell !$ and ${m}_{1}=p-1\text{.}$ We number the vertices from left to right and denote the ${i}^{\text{th}}$ vertex by ${a}_{i}\text{.}$ We must show that $$\sum _{J\subset I}{(-1)}^{J}{p}^{\ell}\ell !/\left|W\left(J\right)\right|={(p-1)}^{\ell}\text{.}$$ Let $S$ be the set of all subsets of the vertex set $I=\{{a}_{1},\dots ,{a}_{\ell}\}\text{.}$ Write $$S=\bigcup _{k=0}^{\ell}{S}_{k}$$ a disjoint union where we put a subset $J\subset I$ in ${S}_{k}$ if $\{{a}_{1},\dots ,{a}_{k}\}\subset J$ and ${a}_{k+1}\notin J\text{.}$ Let ${I}_{k}=\{{a}_{k+2},\dots ,{a}_{\ell}\}$ for $0\le k\le \ell -2$ and put ${I}_{\ell -1}={I}_{\ell}=\varphi \text{.}$ If $J\in {S}_{k}$ then $J=\{{a}_{1},\dots ,{a}_{k}\}\cup H$ where $H\subseteq {I}_{k}$ and thus $$\left|W\left(J\right)\right|={p}^{k}k!\left|W\left(H\right)\right|$$ (Here of course $\left|W\left(\varphi \right)\right|=1\text{).}$ Hence $$\sum _{J\subseteq I}{(-1)}^{J}{\left|W\left(J\right)\right|}^{-1}=\sum _{k=0}^{\ell}\frac{{(-1)}^{k}}{{p}^{k}k!}\sum _{H\subseteq {I}_{k}}{(-1)}^{H}{\left|W\left(H\right)\right|}^{-1}$$ But $W\left({I}_{k}\right)$ is the symmetric group on $\ell -k$ letters, a Coxeter group, so (10) implies $$\sum _{H\subseteq {I}_{k}}{(-1)}^{H}{\left|W\left(H\right)\right|}^{-1}=\frac{1}{(\ell -k)!}$$ Thus $$\begin{array}{ccc}\sum _{J\subseteq I}{(-1)}^{J}|W:W\left(J\right)|& =& \sum _{k=0}^{\ell}{(-1)}^{k}\frac{\ell !}{k!(\ell -k)!}{p}^{\ell -k}\\ & =& {(p-1)}^{\ell}\text{.}\end{array}$$ $\square $ |

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 $\Gamma \in {\mathcal{C}}_{\ell}^{+}$ be connected and put $W=W\left(\Gamma \right)\text{.}$ For $w\in W$ we denote by $\mu \left(w\right)$ the multiplicity of 1 as an eigenvalue of $\theta \left(w\right)$ and we put $k\left(w\right)=\ell -\mu \left(w\right)\text{.}$

Define $\Psi :W\to \u2102$ by $$\Psi \left(w\right)=\sum _{J\subset I}{(-1)}^{J}{1}_{W\left(J\right)}^{W}\left(w\right)$$ where ${1}_{W\left(J\right)}^{W}$ is the character of $W$ induced from the principal character of $W\left(J\right)\text{.}$ We offer the following

Conjecture: Let $w\in W\text{.}$ Then $$\begin{array}{cc}\text{(11)}& \Psi \left(w\right)={(-1)}^{k\left(w\right)}{m}_{1}^{\mu \left(w\right)}\end{array}$$ (Here ${m}_{1}={d}_{1}-1,$ the smallest exponent of $\theta \left(W\right)\text{.)}$

Note that if $W$ is a Coxeter group, ${m}_{1}=1\text{.}$ The non real eigenvalues of $\theta \left(w\right)$ occur in conjugate pairs and the real eigenvalues are $\pm 1\text{.}$ Thus ${(-1)}^{k\left(w\right)}=\text{Det}\left(w\right)$ and hence, in this case, our conjecture is just Theorem 2 of [Sol1966] with $\chi ={1}_{W}\text{.}$

If the graph $\Gamma $ has only one vertex (11) is obvious.

If $\Gamma $ 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 $Z\subset G$ be the subgroup of all those scalar
matrices in $G\text{.}$ Put $\stackrel{\u203e}{G}=G/Z$
and denote the natural map $G\to \stackrel{\u203e}{G}$ by
$g\to \stackrel{\u203e}{g}$ for
$g\in G\text{.}$ Define
$$\begin{array}{ccc}{\u2102}_{G}\left(g\right)& =& \{x\in G:{g}^{x}=g\}\phantom{\rule{2em}{0ex}}\text{and}\\ {\u2102}_{G}^{\prime}\left(g\right)& =& \{x\in G:{g}^{x}=\pm g\}\text{.}\end{array}$$
Then

(a) | If $g\in G$ satisfies $\text{tr}\left(g\right)\ne 0,$ then $${\u2102}_{\stackrel{\u203e}{G}}\left(\stackrel{\u203e}{g}\right)=\stackrel{\u203e}{{\u2102}_{G}\left(g\right)}$$ |

(b) | If $G$ consists of $2\times 2$ matrices, and $g\in G$ satisfies $\text{tr}\left(g\right)=0$ then ${\u2102}_{\stackrel{\u203e}{G}}\left(\stackrel{\u203e}{g}\right)=\stackrel{\u203e}{{\u2102}_{G}^{\prime}\left(g\right)}$ |

Proof. | |

For both (a) and (b) it is clear that the right side is included in the left. So suppose $\stackrel{\u203e}{x}\in {\u2102}_{\stackrel{\u203e}{G}}\left(\stackrel{\u203e}{g}\right)\text{.}$ Hence ${g}^{x}=\lambda g$ some $\lambda \in Z\text{.}$ Taking traces on both sides of this equation we have $$\text{tr}\left(g\right)=\lambda \text{tr}\left(g\right),$$ so $\text{tr}\left(g\right)\ne 0$ implies $\lambda =1\text{.}$ Thus ${g}^{x}=g$ and $\stackrel{\u203e}{x}\in \stackrel{\u203e}{{\u2102}_{G}\left(g\right)},$ yielding (a). In the situation of (b) we take determinants on both sides of the equation ${g}^{x}=\lambda g$ to obtain $$\text{Det}\left(g\right)={\lambda}^{2}\text{Det}\left(g\right)\text{.}$$ Thus ${\lambda}^{2}=1,$ so ${g}^{x}=\pm g$ and $\stackrel{\u203e}{x}\in \stackrel{\u203e}{{\u2102}_{G}^{\prime}\left(g\right)}\text{.}$ $\square $ |

**Corollary 9.** Suppose $\Gamma \in {\mathcal{C}}_{2}^{+}$
is connected; say $\Gamma $ is ${p}_{1}\left[q\right]{p}_{2}\text{.}$
As usual we denote the generators for $W\left(\Gamma \right)$ by
${r}_{1},$ ${r}_{2}$ and let
$G=\theta \left(W\left(\Gamma \right)\right)$
with ${S}_{i}=\theta \left({r}_{i}\right)\text{.}$ Then
$${\u2102}_{W}\left({r}_{i}\right)=\u27e8{r}_{i}\u27e9\times \mathcal{Z}\left(W\right)\text{.}$$

Proof. | |

Suppose ${p}_{i}\ne 2\text{.}$ Since $G$ is irreducible $\mathcal{Z}\left(G\right)$ consists of scalar matrices. Put $\stackrel{\u203e}{G}=G/\mathcal{Z}\left(G\right)\text{.}$ Since ${S}_{i}$ is a reflection ${\stackrel{\u203e}{S}}_{i}$ has order ${p}_{i}\text{.}$ From [STo1954] we know that $\stackrel{\u203e}{G}$ 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 ${\u2102}_{\stackrel{\u203e}{G}}\left({\stackrel{\u203e}{S}}_{i}\right)=\u27e8{\stackrel{\u203e}{S}}_{i}\u27e9\text{.}$ Now ${p}_{i}\ne 2$ further forces $\text{tr}\left({S}_{i}\right)\ne 0,$ and applying Lemma 4(a) we have $\stackrel{\u203e}{{\u2102}_{G}\left({S}_{i}\right)}=\u27e8{\stackrel{\u203e}{S}}_{i}\u27e9\text{.}$ Thus ${\u2102}_{G}\left({S}_{i}\right)=\u27e8{S}_{i}\u27e9\times \mathcal{Z}\left(G\right)\text{.}$ Now suppose ${p}_{i}=2\text{.}$ Consulting List 1 of Theorem 2 we see $\Gamma $ must be one of $3\left[6\right]2,$ $4\left[6\right]2,$ $3\left[8\right]2,$ $5\left[6\right]2,$ $3\left[10\right]2\text{.}$ From [STo1954] we have $2|\left|\mathcal{Z}\left(G\right)\right|$ so that $-I\in \mathcal{Z}\left(G\right)\text{.}$ Hence $-{S}_{2}$ is a reflection in $G\text{.}$ It follows immediately from Theorem 5 that $-{S}_{2}$ is conjugate to ${S}_{2}$ for all of the above graphs with the possible exception of $4\left[6\right]2$ where we must rule out the possibility that $-{S}_{2}$ is conjugate to ${S}_{1}^{2}\text{.}$ In the group $4\left[6\right]2,$ ${S}_{1}{S}_{2}$ has order 24 and $\mathcal{Z}\left(G\right)$ is generated by ${\left({S}_{1}{S}_{2}\right)}^{3},$ an element of order 8. Hence $-I={\left({S}_{1}{S}_{2}\right)}^{12}\text{.}$ If $\lambda $ is a linear character of $G,$ $\lambda \left({S}_{1}\right)={i}^{\alpha}$ and $\lambda \left({S}_{2}\right)={(-1)}^{\beta}$ and thus $\lambda {\left({S}_{1}{S}_{2}\right)}^{12}=1\text{.}$ Thus $-I={\left({S}_{1}{S}_{2}\right)}^{12}\in G\prime ,$ the commutator subgroup of $G\text{.}$ So, $-{S}_{2}$ and ${S}_{2}$ have the same images in $G/G\prime \text{;}$ but a glance at the proof of Lemma 3 reveals that ${S}_{2}$ and ${S}_{1}^{2}$ have distinct images in $G/G\prime \text{.}$ Hence, $-{S}_{2}$ is not conjugate to ${S}_{1}^{2}\text{.}$ So using the notation of Lemma 4 we have $${\u2102}_{G}^{\prime}\left({S}_{2}\right)>{\u2102}_{G}\left({S}_{2}\right)\text{.}$$ In fact, we have $$\mid {\u2102}_{G}^{\prime}\left({S}_{2}\right):{\u2102}_{G}\left({S}_{2}\right)\mid =2\text{.}$$ In light of Lemma 4(b) we see that to show ${\u2102}_{G}\left({S}_{2}\right)=\u27e8{S}_{2}\u27e9\times \mathcal{Z}\left(G\right)$ it suffices to show $$\begin{array}{cc}\text{(12)}& \left|{\u2102}_{\stackrel{\u203e}{G}}\left({\stackrel{\u203e}{S}}_{2}\right)\right|=4\text{.}\end{array}$$ If $\stackrel{\u203e}{G}$ is the alternating group on tour or five letters (12) is obvaous. So we assume $\stackrel{\u203e}{G}$ is the symmetric group on four letters. Consulting [STo1954] we see that $\Gamma $ is $3\left[8\right]2$ or $4\left[6\right]2\text{.}$ If ${\stackrel{\u203e}{S}}_{2}$ is a transposition (12) is again obvious. So assume ${\stackrel{\u203e}{S}}_{2}$ is a product of two disjoint transpositions. Thus ${\stackrel{\u203e}{S}}_{2}$ is an element of the alternating group on four letters. But if $\Gamma $ is $3\left[8\right]2$ then ${\stackrel{\u203e}{S}}_{1},$ an element of order three, is also in the alternating group and hence so is $\stackrel{\u203e}{G}=\u27e8{\stackrel{\u203e}{S}}_{1},{\stackrel{\u203e}{S}}_{2}\u27e9\text{;}$ a contradiction. If $\Gamma $ is $4\left[6\right]2$ then as mentioned before ${\stackrel{\u203e}{S}}_{1}{\stackrel{\u203e}{S}}_{2}$ has order three and hence is in the alternating group. This again is impossible since ${\stackrel{\u203e}{S}}_{1}$ has order four and does not lie in the alternating group. Using Corollary 9 we can now proceed to verify our conjecture (11). If $w\in W,$ $w\ne 1,$ and $\theta \left(w\right)$ is not a reflection, then $w$ is not conjugate to any element of either $\u27e8{r}_{1}\u27e9$ or $\u27e8{r}_{2}\u27e9$ and $\mu \left(w\right)=0$ so $k\left(w\right)=2\text{.}$ Hence $$\Psi \left(w\right)=1={(-1)}^{k\left(w\right)}{m}_{1}^{\mu \left(w\right)}\text{.}$$ If $\theta \left(w\right)$ is a reflection, then by Theorem 5, $w$ is conjugate to an element of $\u27e8{r}_{1}\u27e9$ or an element of $\u27e8{r}_{2}\u27e9\text{.}$ By looking at the matrices one easily sees that if $x\in \u27e8{r}_{i}\u27e9,$ $x\ne 1,$ then ${\u2102}_{w}\left(x\right)={\u2102}_{w}\left({r}_{i}\right)$ and thus by Corollary 9, ${\u2102}_{w}\left(x\right)=\u27e8{r}_{i}\u27e9\times \mathcal{Z}\left(W\right)\text{.}$ Further from [Cox1974] we know that $\left|\mathcal{Z}\left(W\right)\right|\frac{(q,2)h}{q}$ and that ${m}_{1}=\frac{2h}{q}-1\text{.}$ We next separate two cases. Case (i) $q$ is even. Without loss of generality we assume $w$ is to an element of $\u27e8{r}_{1}\u27e9\text{.}$ By Corollary 6 $w$ is not conjugate to an element of $\u27e8{r}_{2}\u27e9\text{.}$ Hence ${1}_{\u27e8{r}_{2}\u27e9}^{W}\left(w\right)=0\text{.}$ Since ${S}_{1},$ ${S}_{1}^{2},$ $\dots ,$ ${S}_{1}^{{p}_{1}-1}$ have distinct non identity eigenvalues the conjugacy classes determined by ${r}_{1},$ ${r}_{1}^{2},$ $\dots ,$ ${r}_{1}^{{p}_{1}-1}$ are all distinct. Hence $$\begin{array}{ccc}{1}_{\u27e8{r}_{1}\u27e9}^{W}\left(w\right)& =& \frac{1}{{p}_{1}}\left|{\u2102}_{W}\left(w\right)\right|\\ & =& \left|\mathcal{Z}\left(W\right)\right|\\ & =& \frac{2h}{q}\end{array}$$ So $\Psi \left(w\right)=1-\frac{2h}{q}=-{m}_{1}={(-1)}^{k\left(w\right)}{m}_{1}^{\mu \left(w\right)}\text{.}$ Case (ii) $q$ is odd. Thus, ${p}_{1}={p}_{2}=p,$ say and ${r}_{1}$ is conjugate to ${r}_{2}\text{;}$ so $w$ is conjugate to an element of $\u27e8{r}_{1}\u27e9$ and to an element of $\u27e8{r}_{2}\u27e9\text{.}$ Hence $${1}_{\u27e8{r}_{1}\u27e9}^{W}\left(w\right)={1}_{\u27e8{r}_{2}\u27e9}^{W}\left(w\right)=\frac{1}{p}\left|{\u2102}_{W}\left(w\right)\right|=\frac{h}{q},$$ So again, $\Psi \left(w\right)=1-\frac{2h}{q}=-{m}_{1}={(-1)}^{k\left(w\right)}{m}_{1}^{\mu \left(w\right)}\text{.}$ This completes the verification of our conjecture in case $\ell =2\text{.}$ $\square $ |

Let $\Gamma \in {\mathcal{C}}_{\ell}^{+}\text{.}$ Put $W=W\left(\Gamma \right)$ and let $I$ denote the vertex set of $\Gamma \text{.}$ Let ${B}_{i}=\{w\in W\hspace{0.17em}|\hspace{0.17em}\mu \left(w\right)=\ell -1\}$ $0\le i\le \ell $ and put ${b}_{i}=\left|{B}_{i}\right|\text{.}$ As usual we let ${m}_{i}={d}_{i}-1$ where ${d}_{1}-1\le \dots \le {d}_{\ell}-1$ are the exponents of $\theta \left(W\right)\text{.}$ In [STo1954] Shephard and Todd introduced and verified and later in [Sol1963] Solomon gave a proof for the polynomial identity: $$\begin{array}{cc}\text{(13)}& \sum _{i=0}^{\ell}{b}_{i}{t}^{i}=\prod _{i=1}^{\ell}(1+{m}_{i}t)\text{.}\end{array}$$

For $J\subset I$ let $V\left(J\right)\subset V$ be the subspace of $V$ spanned by the basis vectors corresponding to the vertices in $J\text{.}$ Also let ${m}_{1J}\le {m}_{2J}\le \dots \le {m}_{\left|J\right|J}$ be the exponents of $\theta \left(W\left(J\right)\right)$ acting on $V\left(J\right)\text{.}$ Further defining ${\mu}_{J}$ on $W\left(J\right)$ as we defined $\mu $ on $W$ and putting ${k}_{J}=\left|J\right|-{\mu}_{J}$ one easily sees that if $x\in W\left(J\right)$ then $k\left(x\right)={k}_{J}\left(x\right)\text{.}$ Finally define a class function $\tau :W\to \u2102\left[t\right]$ by $\tau \left(w\right)={t}^{k\left(w\right)}\text{.}$ Notice that the left hand side of (13) can be written as $\sum _{w\in W}\tau \left(w\right)\text{.}$

Consider the sum $$\begin{array}{ccc}\sum _{w\in W}\Psi \left(w\right)\tau \left(w\right)& =& \sum _{w\in W}\tau \left(w\right)\sum _{J\subset I}{(-1)}^{J}{1}_{W\left(J\right)}^{W}\left(w\right)\\ & =& \sum _{J\subset I}{(-1)}^{J}\sum _{w\in W}{1}_{W\left(J\right)}^{W}\left(w\right)\tau \left(w\right)\\ & =& \sum _{J\subset I}{(-1)}^{J}\left|W\right|[{1}_{W\left(J\right)}^{W},\tau ]\\ & =& \sum _{J\subset I}{(-1)}^{J}\left|W\right|[{1}_{W\left(J\right)},{\tau}_{W\left(J\right)}]\\ & =& \sum _{J\subset I}{(-1)}^{J}|W:W\left(J\right)|\sum _{w\in W\left(J\right)}\tau \left(w\right)\\ & =& \sum _{J\subset I}{(-1)}^{J}|W:W\left(J\right)|\prod _{i=1}^{\left|J\right|}(1+{m}_{ij}t)\end{array}$$ In the last step we have used (13) applied to the groups $W\left(J\right)\text{.}$

Next consider the sum $$\begin{array}{ccc}\sum _{w\in W}{(-1)}^{k\left(w\right)}{m}_{1}^{\mu \left(w\right)}{t}^{k\left(2\right)}& =& \sum _{w\in W}{(-1)}^{k\left(w\right)}{m}_{1}^{\ell -k\left(w\right)}{t}^{k\left(w\right)}\\ & =& {m}_{1}^{\ell}\sum _{w\in W}{(-\frac{t}{{m}_{1}})}^{k\left(w\right)}\\ & =& {m}_{1}^{\ell}\prod _{i=1}^{\ell}(1-\frac{{m}_{i}}{{m}_{1}}t)\\ & =& \prod _{i=1}^{\ell}({m}_{1}-{m}_{i}t)\end{array}$$ 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: $$\begin{array}{cc}\text{(14)}& \sum _{J\subset I}{(-1)}^{J}|W:W\left(J\right)|\prod _{i=1}^{\left|J\right|}(1+{m}_{ij}t)=\prod _{i=1}^{\ell}({m}_{1}-{m}_{i}t)\text{.}\end{array}$$ In case $W$ is a Coxeter group, ${m}_{1}=1$ 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 ${\mathcal{C}}_{\ell}^{+}$ with the exception of the infinite family ${B}_{\ell}^{p}\text{.}$ This lends another measure of credibility to our conjecture (11).

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.