## Discrete complex reflection groups

Last update: 12 May 2014

## Notes and References

This is an excerpt of the lecture notes Discrete complex reflection groups by V.L. Popov. Lectures delivered at the Mathematical Institute, Rijksuniversiteit Utrecht, October 1980.

## The structure of $r\text{-groups}$ in the case $s=n+1$

When $s=n+1$ it is no longer true in general that an infinite irreducible complex crystallograph $r\text{-group}$ $W$ is the semidirect product of $\text{Lin} W$ and $\text{Tran} W\text{.}$ We shall explain here how one can find the corresponding extensions of $\text{Tran} W$ by $\text{Lin} W$ in this case.

We assume that $k=ℂ\text{.}$

### The Cocycle $c$

We recall that the structure of the extension is given by a $1\text{-cocycle}$ $c$ of $\stackrel{\sim }{K},$ where $K=\text{Lin} W,$ with values in $V\text{;}$ see Section 2.4.

Let $\mathrm{\Gamma }=\text{Tran} W$ and $a\in E\text{.}$ Taking $a$ as an origin, we can identify $A\left(E\right)$ and $GL\left(V\right)V\text{;}$ then $W$ consists of the elements $\left(P,c\left(P\right)+\mathrm{\Gamma }\right),$ $P\in K\text{.}$

First of all, one can assume, upon replacing $c$ by a suitable cocycle homologuous to $c,$ that $c(r1)=…= c(rn)=0.$ We assume that the order of ${R}_{j}$ is equal to $m\left({H}_{{R}_{j}}\right),$ $1\le j\le s,$ see Section 3.2. Proof. Let ${\gamma }_{1},\dots ,{\gamma }_{n}\in W$ be the reflections with $\text{Lin} {\gamma }_{j}={R}_{j},$ $1\le j\le n\text{.}$ We assume, as usual, that the group $K\prime$ generated by ${R}_{j},$ $1\le j\le n,$ is irreducible. We have $\underset{j=1}{\overset{n}{\cap }}{H}_{{R}_{j}}=0,$ hence $\underset{j=1}{\overset{n}{\cap }}{H}_{{\gamma }_{j}}=b\in E\text{.}$ Then we have $κb(γj)= (Rj,0), j =1,…,n,$ and we are done. $\square$

We shall assume now that $c(r1)=…= c(rn)=0.$

Therefore $c$ is defined by only one vector $c\left({r}_{n+1}\right)\text{.}$ Moreover, one can take $c(rn+1)=λ en+1, λ∈ℂ$ because there exists a reflection $\left({R}_{n+1},v\right)$ in $W\text{.}$

Therefore, given an invariant lattice $\mathrm{\Gamma }$ of full rank, one has to solve the following problems

Find those $\lambda \in ℂ$ for which

 a) the cocycle $c$ of $\stackrel{\sim }{K}$ with values in $V,$ given by condition $c(rj)=0, 1≤j≤n; c (rn+1)=λ en+1, λ∈ℂ,$ satisfies $c(F)∈Γ$ for any relation $F$ of $K$ (i.e. for every element $F$ of $\text{Ker} \varphi ,$ see Section 2.4). b) Condition a) holds and the corresponding group $W$ is an $r\text{-group.}$

Theorem.

 1) If $\mathrm{\Gamma }={\mathrm{\Gamma }}^{0}$ then a) $⇒$ b). 2) If $c\left(F\right)\in {\mathrm{\Gamma }}^{0}$ for every relation $F$ then b) $⇒$ $\mathrm{\Gamma }={\mathrm{\Gamma }}^{0}\text{.}$ Proof. 1) Let $\mathrm{\Gamma }={\mathrm{\Gamma }}^{0}\text{.}$ We know that $Γ0=Γ1+…+ Γn+1.$ But $\mathrm{\Gamma }\prime ={\mathrm{\Gamma }}_{1}+\dots +{\mathrm{\Gamma }}_{n}$ is a root lattice for $K\prime$ and the condition $c\left({r}_{1}\right)=\dots =c\left({r}_{n}\right)=0$ shows that the semidirect product of $K\prime$ and $\mathrm{\Gamma }\prime$ lies in $W\text{.}$ This semidirect product is an $r\text{-group,}$ see Section 4.5. We also have that $W$ contains the reflections $\left({R}_{n+1},\lambda {e}_{n+1}+t\right),$ $t\in {\mathrm{\Gamma }}_{n+1}\text{.}$ Let $\gamma \in W$ be an arbitrary element, $\gamma =\left(P,v\right)\text{.}$ Then $P$ is a product of reflections ${R}_{j},$ $1\le j\le n+1,$ in some order. Therefore, multiplying $\gamma$ by $\left({R}_{j},0\right),$ $1\le j\le n,$ and by $\left({R}_{n+1},\lambda {e}_{n+1}\right)$ in a suitable order, one can obtain an element of the form $\left(1,t\right)\text{.}$ But the elements $\left(1,t\prime \right),$ $t\prime \in \mathrm{\Gamma }\prime ,$ are also products of reflections. Therefore, multiplying $\gamma$ by reflections, one can obtain $\left(1,r\right),$ $r\in {\mathrm{\Gamma }}_{n+1}\text{.}$ But $(Rn+1,λen+1) (Rn+1,λen+1+r) (1,r)=(1,0).$ Therefore $\gamma$ is a product of reflections. 2) We have $c\left(F\right)\in {\mathrm{\Gamma }}^{0}\text{.}$ Therefore $c,$ in fact, defines a $1\text{-cocycle}$ of $K$ with values in $V/{\mathrm{\Gamma }}^{0}\text{.}$ Let $W\prime$ be the group defined by $c,$ with $\text{Lin} W\prime =K$ and $\text{Tran} W\prime ={\mathrm{\Gamma }}^{0}\text{.}$ It is an $r\text{-group}$ because of 1). Also we have a group $W$ defined by $c,$ with $\text{Lin} W=K$ and $\text{Tran} W=\mathrm{\Gamma }\text{.}$ Let $\gamma \in W$ be a reflection. By section 3.2, we know, there exists $\delta \in W\prime$ with $\delta \gamma {\delta }^{-1}=\left({R}_{j}^{l},t\right)$ for certain $l$ and $j\text{.}$ This is also a reflection, hence $t\perp {H}_{{R}_{j}}\text{.}$ But $t=c\left({r}_{j}^{l}\right)+v,$ $v\in \mathrm{\Gamma }\text{.}$ We have $c(rjl)= (1+rj+rj2+…+rjl-1) c(rj).$ By definition of $c,$ we have $c\left({r}_{j}\right)\perp {H}_{{R}_{j}}\text{.}$ Hence, also $c\left({r}_{j}^{l}\right)\perp {H}_{{R}_{j}}\text{.}$ Therefore, $v\perp {H}_{{R}_{j}},$ or, in other words, $v\in {\mathrm{\Gamma }}^{0}\text{.}$ This means that $\delta \gamma {\delta }^{-1}\in W\prime \text{.}$ It follows now from $\delta \in W\prime$ that $\gamma \in W\prime \text{;}$ hence $W=W\prime$ and $\mathrm{\Gamma }={\mathrm{\Gamma }}^{0}\text{.}$ $\square$

The following simple observation is very useful in practice because it gives strong restrictions on the choice of $\lambda \text{:}$

Theorem. Let a) be satisfied and let $P\in K\prime$ be such that $Rn+1P Rn+1-1 ∈K′.$ Then $λ(1-Rn+1PRn+1-1) en+1∈Γ.$ Proof. It follows from the definition of $c$ that $\left(P,0\right)$ and $\left({R}_{n+1},\lambda {e}_{n+1}\right)\in W\text{.}$ We have now: $(Rn+1,λen+1) (P,0) (Rn+1,λen+1)-1= ( Rn+1P Rn+1-1, -Rn+1P Rn+1-1 (λen+1) +λen+1 )$ and we are done. $\square$

### Example

$K={K}_{31}\text{.}$ The graph is $1 2 2 2 2 3 4 5 i i-1$ The vectors ${e}_{1},{e}_{2},\dots ,{e}_{5}$ are given in Table 2. Note that ${e}_{5}=i{e}_{1}+{e}_{2}+{e}_{3}\text{.}$

There exists only one (up to similarity) $K\text{-invariant}$ lattice $\mathrm{\Gamma }$ of full rank $Γ=[1,i]e1+ …+[1,i]e4.$

The presentation (i.e. the sets of generators and relations) of $K$ is known, see [Cox1974].

The relations are: $R12=R22= R32=(R2R3)3 =(R3R1)3= (R1R2)3=1, (R2R1R3R1)4=1, (R4R5)3=1. R52=(R5R2)2= (R5R1R3R1)2= (R5R3)4=R1 (R5R3R2R3)R1 (R5R3R2R3)-1=1 R42=(R4R1)2 =(R4R3)2= (R4R2)3=1$ It follows from ${R}_{1}\left({R}_{5}{R}_{3}{R}_{2}{R}_{3}\right){R}_{1}{\left({R}_{5}{R}_{3}{R}_{2}{R}_{3}\right)}^{-1}={R}_{5}^{2}=1$ that $R5R1R5∈K′.$

Therefore $Γ∋λ (R1R5R1R5) e5=λ ((1+i)e1+2e2+2e3)$ and hence $\lambda =\frac{a+bi}{2},$ $a,b\in ℤ,$ and $a\equiv b$ (mod $2\text{).}$ But $\left[1,i\right]{e}_{5}\in \mathrm{\Gamma }\text{.}$ Therefore, we may assume that $λ=1+i2.$

It only remains to check whether $c\left(F\right)\in \mathrm{\Gamma }$ or not for this particular $\lambda \text{.}$ This has to be done by straightforward computations: $c(r52) = c(r5)+r5c (r5)=1+i2 (1+r5)e=0∈Γ. c((r4r5)3) = (1+r4r5+(r4r5)2) (c(r4)+r4c(r5)) = 1+i2 (1+r4r5+(r4r5)2) (r4e5)=0∈Γ,$ and so on (one only needs to consider the relations which involve ${R}_{5}\text{).}$ This check shows that $\lambda =\frac{1+i}{2}$ gives a cocycle of $K,$ indeed, and, hence, defines an $r\text{-group}$ $W$ with $\text{Lin} W=K,$ $\text{Tran} W=\mathrm{\Gamma }\text{.}$ It can be proved in a straightforward manner that this cocycle $c$ is not a coboundary, i.e. $W$ is not a semidirect product.

The same considerations can be given for each $K$ with $s=n+1,$ and, as a result, one obtains Table 2. It appears a posteriori that in all cases either $\mathrm{\Gamma }$ is a root lattice or $c\left(F\right)\in {\mathrm{\Gamma }}^{0}$ for every relation $F$ of $K$ (therefore, for an $r\text{-group}$ $W,$ $\text{Tran} W$ is always a root lattice). By performing similar straightforward computations in the remaining cases, a proof of the theorems in Section 2.8. is obtained.

As for the proof of the theorem from Section 2.9, the first part of it is given in Section 4.6; the second part (about minimality of $ℤ\left[\text{Tr}K\right]\text{)}$ follows from [BSh1963]. The third part follows from Table 1.