## Complex Reflection Groups

Last update: 10 January 2014

## Section III

Let $\Gamma \in {𝒞}_{\ell }$ and put $R=\left\{{r}_{1},\dots ,{r}_{\ell }\right\}$ the set of generators for $W=W\left(\Gamma \right)\text{.}$ We define an equivalence relation $\text{"}\sim \text{"}$ on $R$ as follows. We say ${r}_{i}\sim {r}_{i}$ and if $i\ne j$ we say ${r}_{i}\sim {r}_{j}$ if and only if there is a sequence of vertices of $\Gamma ,$ ${a}_{i}={a}_{{i}_{1}},{a}_{{i}_{2}},\dots ,{a}_{{i}_{m}}={a}_{j},$ such that ${q}_{{i}_{k}{i}_{k+1}}$ is odd for all $1\le k\le m-1\text{.}$ We then write $R={R}^{1}\cup {R}^{2}\cup \dots \cup {R}^{n}$ as the disjoint union of equivalence classes. Since ${r}_{i}$ is conjugate to ${r}_{j}$ if ${q}_{ij}$ is odd all the elements of an equivalence class ${R}^{k}$ are conjugate and we denote by ${p}_{{i}_{k}}$ the integer which is the common label on all those vertices in $\Gamma$ corresponding to the generators in ${R}^{k}\text{.}$

Lemma 3. Let $\Gamma \in {𝒞}_{\ell }$ and let $W=W\left(\Gamma \right)\text{.}$ Let $W\prime$ denote the commutator subgroup of $W\text{.}$ Then using the notation just introduced above we have $|W/W′|= ∏k=1npik.$

 Proof. We denote the natural map $W\to W/W\prime$ by $x\to \stackrel{‾}{x}\text{.}$ Then since $W$ is generated by ${r}_{1},\dots ,{r}_{\ell },$ $\stackrel{‾}{W}$ is generated by ${\stackrel{‾}{r}}_{1},\dots ,{\stackrel{‾}{r}}_{\ell }\text{.}$ But in the map $W\to W/W\prime$ all the elements of an equivalence class ${R}^{k}$ are identified. Thus $W/W\prime$ is generated by $n$ elements (perhaps not all different) of orders at most ${p}_{{i}_{1}},\dots ,{p}_{{i}_{n}}\text{.}$ Hence $|\stackrel{‾}{W}|\le \prod _{k=1}^{n}{p}_{{i}_{k}}\text{.}$ Now let $A=⟨{\epsilon }_{{i}_{1}}⟩×\dots ×⟨{\epsilon }_{{i}_{n}}⟩$ the direct product of $n$ cyclic groups of orders ${p}_{{i}_{1}},\dots ,{p}_{{i}_{n}}\text{.}$ We define $\gamma :R\to A$ by $γ(rj)= ( 1,…,1, εik ⏟kth position ,1,…,1 )$ if ${r}_{j}\in {R}^{k}\text{.}$ We claim that $\gamma$ can be extended to a homomorphism of $W$ onto $A\text{.}$ For clearly we have ${\left(\gamma \left({r}_{j}\right)\right)}^{{p}_{j}}=1\text{.}$ Further if the ${j}^{\text{th}}$ and ${j\prime }^{\text{th}}$ vertices of $\Gamma$ are not joined by an edge, then $\gamma \left({r}_{j}\right)$ and $\gamma \left({r}_{j\prime }\right)$ commute as $A$ is abelian. Now if ${q}_{j,j\prime }$ is odd, then ${r}_{j}$ and ${r}_{j\prime }$ are in the same equivalence class and hence $\gamma \left({r}_{j}\right)=\gamma \left({r}_{j\prime }\right)$ so the relation $\gamma \left({r}_{j}\right)\gamma \left({r}_{j\prime }\right)\gamma \left({r}_{j}\right)\dots =\gamma \left({r}_{j\prime }\right)\gamma \left({r}_{j}\right)\gamma \left({r}_{j\prime }\right)\dots$ with ${q}_{jj\prime }$ factors on each side is obviously satisfied. Finally if ${q}_{jj\prime }$ is even the relation $\gamma \left({r}_{j}\right)\gamma \left({r}_{j\prime }\right)\gamma \left({r}_{j}\right)\dots =\gamma \left({r}_{j\prime }\right)\gamma \left({r}_{j}\right)\gamma \left({r}_{j\prime }\right)\dots$ with ${q}_{jj\prime }$ factors on each side becomes $( γ(rj) γ(rj′) ) qjj′2 = ( γ(rj′) γ(rj) ) qjj′2$ which is obvious as $A$ is abelian. So we have our claim and thus $|W‾|≥|A| =∏k=1npik.$ $\square$

Corollary 6. Let $\Gamma \in 𝒞$ and let $W=W\left(\Gamma \right)\text{.}$ If ${r}_{i}$ and ${r}_{j}$ are in distinct equivalence classes under the equivalence relation $\text{"}\sim \text{"},$ then no non-identity element of $⟨{r}_{i}⟩$ is conjugate in $W$ to any element of $⟨{r}_{j}⟩\text{.}$

Let $V$ be a finite dimensional complex vector space and let $G$ be a finite subgroup of $GL\left(V\right)\text{.}$ We say $R\in G$ is a reflection if $R\ne 1$ and if there is a hyperplane $U\subset V$ fixed by $R\text{.}$ $U$ is called the reflecting hyperplane for the reflection $R\text{.}$ We let $𝒰=𝒰\left(G\right)$ be the collection of hyperplanes $U$ in $V$ such that $U$ is the reflecting hyperplane for some reflection $R\in G\text{.}$ Note that if $U\in 𝒰$ is the reflecting hyperplane for $R\in G$ and if $T\in G,$ then $T\left(U\right)\in 𝒰$ as it is the reflecting hyperplane for the reflection $TR{T}^{-1}\text{.}$ Thus $G$ acts on $𝒰\text{.}$ For $U\in 𝒰$ we let $C\left(U\right)$ be the subgroup of $G$ consisting of all elements of $G$ which are the identity on $U\text{.}$ Since $G$ is finite each $C\left(U\right),$ for $U\in 𝒰,$ is isomorphic to a finite subgroup of $ℂ\\left\{0\right\}$ and hence is a cyclic group. We denote its order by $e\left(U\right)\text{.}$ Then if $\Delta$ is an orbit of $G$ on $𝒰$ and if $U,U\prime \in \Delta$ we have $C\left(U\right)$ and $C\left(U\prime \right)$ are conjugate, so $e\left(U\right)=e\left(U\prime \right)\text{.}$ We thus denote by $e\left(\Delta \right)$ the common value of $e\left(U\right)$ for $U\in \Delta \text{.}$

The following result appears in [Spr1974].

Proposition 9. Let $\Gamma \in {𝒞}^{+}$ and let $G=\theta \left(W\left(\Gamma \right)\right)\text{.}$ Let $G\prime$ denote the commutator subgroup of $G\text{.}$ Then $G/G\prime$ is the direct product of cyclic groups of orders $e\left(\Delta \right)$ as $\Delta$ runs through the orbits of $G$ on $𝒰\left(G\right)\text{.}$ In particular $|G/G\prime |=\prod _{\Delta }e\left(\Delta \right)\text{.}$

Since for $\Gamma \in {𝒞}^{+},$ $\theta$ is a faithful representation of $W\left(\Gamma \right),$ Proposition 9 gives another way of computing the order of $W/W\prime \text{.}$ By using both these methods and comparing results we can give a case by case verification of the following theorem which is a known result for Coxeter groups [Bou1968, p.74].

Theorem 5. Let $\Gamma \in {𝒞}_{\ell }^{+}$ and let $G=\theta \left(W\left(\Gamma \right)\right)\text{.}$ Then every reflection in $G$ is conjugate in $G$ to a power of one of the generating reflections, ${S}_{1},\dots ,{S}_{\ell }\text{.}$

Proof.

We may assume $\Gamma$ is connected. Thus $\Gamma$ occurs in List 1. Now if $\Gamma$ has just one vertex there is nothing to prove. So we consider the case where $\Gamma$ has two vertices. Say $\Gamma$ is ${p}_{1}\left[q\right]{p}_{2}\text{.}$ For $i=1,2,$ we put ${U}_{i}=$ the reflecting hyperplane for ${S}_{i},$ ${\Delta }_{i}=$ the orbit containing ${U}_{i},$ and ${e}_{i}=e\left({\Delta }_{i}\right)\text{.}$ Since ${S}_{i}\in C\left({U}_{i}\right)$ we have ${p}_{i}|{e}_{i}$ so in particular ${p}_{i}\le {e}_{i}\text{.}$

Now if $q$ is odd we apply Lemma 3 to obtain $|G/G\prime |={p}_{1}\text{.}$ Now clearly Proposition 9 implies $|G/G\prime |\ge {e}_{1}\text{.}$ We thus obtain the sequence $|G/G\prime |\ge {e}_{1}\ge {p}_{1}=|G/G\prime |\text{.}$ Hence ${e}_{1}={p}_{1}$ and ${\Delta }_{1}$ is the only orbit of $G$ on $𝒰\left(G\right)\text{.}$ Thus every reflection in $G$ is conjugate in $G$ to a power of ${S}_{1}\text{.}$

If $q$ is even we apply Lemma 3 to obtain $|G/G\prime |={p}_{1}{p}_{2}\text{.}$ Now if ${\Delta }_{1}\ne {\Delta }_{2}$ we can apply Proposition 9 to get $|G/G\prime |\ge {e}_{1}{e}_{2}\text{.}$ Thus if ${\Delta }_{1}\ne {\Delta }_{2}$ we have the sequence $|G/G\prime |\ge {e}_{1}{e}_{2}\ge {p}_{1}{p}_{2}=|G/G\prime |$ which forces ${p}_{i}={e}_{i}$ implying the desired result. Thus we must show ${\Delta }_{1}\ne {\Delta }_{2}\text{.}$ Assume to the contrary that ${\Delta }_{1}={\Delta }_{2}\text{.}$ Then $C\left({U}_{1}\right)$ is conjugate in $G$ to $C\left({U}_{2}\right)\text{.}$ Let $d=\left({p}_{1},{p}_{2}\right)\text{.}$ If $d\ne 1,$ $⟨{S}_{i}⟩$ has a subgroup ${D}_{i}$ of order $d>1$ $\left(i=1,2\right)\text{.}$ Now ${D}_{1}\subset C\left({U}_{1}\right)$ and ${D}_{2}\subset C\left({U}_{2}\right)$ and thus ${D}_{1}$ is conjugate in $G$ to ${D}_{2}$ since $C\left({U}_{1}\right)$ and $C\left({U}_{2}\right)$ are cyclic and conjugate. Thus $⟨{r}_{1}⟩$ has a non trivial subgroup which is conjugate in $W\left(\Gamma \right)$ to a subgroup of $⟨{r}_{2}⟩\text{.}$ This is a contradiction to Corollary 6. Hence we are forced to assume $d=1\text{.}$ Thus ${p}_{1}{p}_{2}|{e}_{1}$ and we have the existence of a reflection $R\in C\left({U}_{1}\right)$ of order ${p}_{1}{p}_{2}\text{.}$ Now $Z\left(G\right)$ consists of scalar matrices as $G$ is irreducible. Thus $R$ also has order ${p}_{1}{p}_{2}$ when viewed as an element in $G/Z\left(G\right)\text{.}$ But $G/Z\left(G\right)$ is isomorphic to the alternating group on four letters, the symmetric group on four letters, or the alternating group on five letters [STo1954, p.279]. Now none of these groups contain an element of order higher than five. But $\left({p}_{1},{p}_{2}\right)=1$ forces ${p}_{1}{p}_{2}\ge 6\text{.}$ So we have the theorem for graphs with two vertices.

If $\Gamma$ has three vertices but is not in any of the infinite families in List 1, we see it is one of

 (1) $2\left[3\right]2\left[5\right]2$ (2) $3\left[3\right]3\left[3\right]3$ (3) $3\left[3\right]3\left[4\right]2$
We number the vertices from left to right and let ${U}_{i}=$ the reflecting hyperplane for ${S}_{i},$ ${\Delta }_{i}=$ the orbit of ${U}_{i},$ and ${e}_{i}=e\left({\Delta }_{i}\right)\text{.}$ Again we remark that $⟨{S}_{i}⟩\subset C\left({U}_{i}\right)$ and thus ${p}_{i}|{e}_{i}$ so in particular, ${p}_{i}\le {e}_{i}\text{.}$

For (1) we apply Lemma 3 to get $|G/G\prime |=2\text{.}$ Proposition 9 certainly implies that $|G/G\prime |\ge {e}_{1}\text{.}$ We thus obtain the sequence $|G/G′|≥e1≥ p1=2= |G/G′|.$ Hence, ${e}_{1}={p}_{1}$ and ${\Delta }_{1}$ is the only orbit.

For (2) the argument is the same as for (1).

For (3) we apply Lemma 3 to obtain $|G/G\prime |=6\text{.}$ Thus if ${\Delta }_{1}\ne {\Delta }_{3}$ we can apply Proposition 9 as we have before to obtain the result. If ${\Delta }_{1}={\Delta }_{3},$ there is a reflection $R\in G$ of order $6\text{.}$ So the eigenvalues of R are $1,1,\delta$ where $\delta$ is a primitive sixth root of unity. We obtain a contradiction as follows:

$SL\left(2,3\right)$ acts naturally on a two dimensional vector space $P$ over $GF\left(3\right)\text{.}$ Viewing the vector space as an abelian group of row vectors we can form the semi-direct product $K=P⋊SL\left(2,3\right)\text{.}$ $K$ can be viewed as the matrix group ${ ( A 00 xy 1 ) | A∈SL(2,3) x,y∈GF(3) } .$ We note that $|K|=216\text{.}$ We define a homomorphism $G\to K,$ denoted by $x\to \stackrel{‾}{x},$ by putting $S‾1= ( 110 010 001 ) , S‾2= ( 100 -110 001 ) ,S‾3= ( -100 0-10 010 ) .$ It is easy to see that $\stackrel{‾}{G}=K\text{.}$ Further ${\left({\stackrel{‾}{S}}_{1}{\stackrel{‾}{S}}_{2}{\stackrel{‾}{S}}_{3}\right)}^{3}={1}_{K}$ and in $G$ we compute ${\left({S}_{1}{S}_{2}{S}_{3}\right)}^{3}=-{\epsilon }_{1}^{2}{I}_{3}$ a central element of order six. Now comparing, $\left[3\right]$ with $\left[2\right]$ we see that $|G|=1296,$ $|Z\left(G\right)|=6\text{.}$ Hence the kernel of the homomorphism defined above is $Z\left(G\right)\text{.}$ Hence, since $R$ is a reflection, $|R|=|\stackrel{‾}{R}|\text{.}$ Now it is easy to see that there are exactly two conjugacy classes of elements of order six in $K$ -- one determined by $\stackrel{‾}{{S}_{1}{S}_{3}}$ and the other by its inverse. Thus, in $G,$ $R$ must be conjugate to an element of $Z\left(G\right){S}_{1}{S}_{3}$ or of $Z\left(G\right){\left({S}_{1}{S}_{3}\right)}^{-1}\text{.}$ Now the eigenvalues of ${S}_{1}{S}_{3}$ are $1,$ $-1,$ ${\epsilon }_{1}$ and those of ${\left({S}_{1}{S}_{3}\right)}^{-1}$ are $1,$ $-1,$ ${\epsilon }_{1}^{2}\text{.}$ In particular for either ${S}_{1}{S}_{3}$ or ${\left({S}_{1}{S}_{3}\right)}^{-1}$ the eigenvalues are distinct. Hence the same is true for any scalar multiple of either ${S}_{1}{S}_{3}$ or ${\left({S}_{1}{S}_{3}\right)}^{-1}\text{.}$ Thus the element $R,$ with eigenvalues $1,1,\delta ,$ cannot be conjugate to such a scalar multiple.

If $\Gamma$ has four vertices but is not in one of the infinite families in List 1 it is either ${H}_{4},$ ${F}_{4},$ or $3\left[3\right]3\left[3\right]3\left[3\right]3\text{.}$ The first two are Coxeter groups for which the result is known. For the last graph we number the vertices from left to right and use the familiar notation ${U}_{i},$ ${\Delta }_{i},$ ${e}_{i}\text{.}$ We apply Lemma 3 to obtain the equality $|G/G\prime |=3\text{.}$ Now Proposition 9 gives us the inequality $|G/G\prime |\ge {e}_{1}\text{.}$ Putting them together with the fact that ${e}_{1}\ge {p}_{1}$ we have the sequence $|G/G′|≥e1≥ p1=3=|G/G′|.$ Thus ${e}_{1}={p}_{1}$ and ${\Delta }_{1}$ is the only orbit.

Now all of ${E}_{6},$ ${E}_{7},$ ${E}_{8},$ ${A}_{\ell },$ ${D}_{\ell }$ are graphs of Coxeter groups for which the result is known. Or, just as easily for all these groups, an application of Lemma 3 gives $|G/G\prime |=2$ and thus applying Proposition 9 yields the result. The only remaining case is ${B}_{\ell }^{p}\text{.}$ Applying Lemma 3 gives $|G/G\prime |=2p\text{.}$ So numbering from the left as usual we see that if ${\Delta }_{1}\ne {\Delta }_{2}$ an application of Proposition 9 will again yield the result. Now if $p$ is even we can argue just as we did for the graph ${p}_{1}\left[q\right]{p}_{2}$ with $q$ even and $\left({p}_{1},{p}_{2}\right)\ne 1$ to show that ${\Delta }_{1}\ne {\Delta }_{2}\text{.}$ So we assume $p$ is odd. Then ${\Delta }_{1}={\Delta }_{2}$ forces the existence of a reflection in $C\left({U}_{1}\right)$ of order $2p\text{.}$ In fact, by Proposition 9, in this situation we must have $|C\left({U}_{1}\right)|=2p$ and ${\Delta }_{1}$ is the only orbit. But in the basis $\left\{{x}_{1},\dots ,{x}_{\ell }\right\}$ defined by ${x}_{i}=\frac{1}{\alpha }{v}_{1}+{v}_{2}+\cdots +{v}_{i}$ $\left(\alpha ={\left(2\text{sin} \pi /p\right)}^{\frac{1}{2}}\right)$ the matrices for the generators ${S}_{k}$ are the standard generators for the full monomial group of ${p}^{\ell }\ell !$ Namely ${S}_{1}\left({x}_{1}\right)={\epsilon }_{1}{x}_{1},$ ${S}_{1}\left({x}_{j}\right)={x}_{j}$ for $j\ne 1\text{.}$ If $k\ne 1,$ ${S}_{k}$ interchanges ${x}_{k}$ and ${x}_{k-1}$ and fixes the other basis vectors. In this basis it becomes clear that $C\left({U}_{1}\right)=⟨{S}_{1}⟩$ a cyclic group of order $p\text{.}$ Thus, ${\Delta }_{1}\ne {\Delta }_{2}\text{.}$

$\square$

## 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.