Reflection Group Examples

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

Last updates: 20 October 2010

The groups $T$, $O$ and $I$

The rank $2$ exceptional complex reflection groups $G_4,\dots ,G_{22}$ in the list of Shephard and Todd, are arll built from the $4$ basic groups, $$I_2\left(4\right)=\text{the dihedral group of order 8.}$$ The tetrahedral group $$T=\text{the tetrahedral group (order 24)}\cong S_4$$ is generated by the matrices $$S_1=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),T_1=\frac{1}{\sqrt[\mathrm{}]{2}}\left(\begin{array}{cc}\epsilon & {\epsilon }^3\\ \epsilon & {\epsilon }^7\end{array}\right),S_1{T}_1=\frac{1}{\sqrt[\mathrm{}]{2}}\left(\begin{array}{cc}{\epsilon }^3& {\epsilon }^5\\ {\epsilon }^7& {\epsilon }^5\end{array}\right)$$ where $i=e^{2\pi i/4}$ and $\epsilon =e^{2\pi i/8}$. These matrices satisfy the relations $$S_1^2=T_1^3=-1,{\left(S_1{T}_1\right)}^3=1.$$ The octahedral group, $$O=\text{the octahedral group (order 48)}\cong W{B}_3,$$ is generated by the matrices $$S_1=\frac{1}{\sqrt[\mathrm{}]{2}}\left(\begin{array}{cc}i& 1\\ -1& -i\end{array}\right),T_1=\frac{1}{\sqrt[\mathrm{}]{2}}\left(\begin{array}{cc}\epsilon & \epsilon \\ {\epsilon }^3& {\epsilon }^7\end{array}\right),S_1{T}_1=\left(\begin{array}{cc}{\epsilon }^3& 0\\ 0& {\epsilon }^5\end{array}\right)$$ where $i=e^{2\pi i/4}$ and $\epsilon =e^{2\pi i/8}$. These matrices satisfy the relations $$S_1^2=T_1^3={\left(S_1{T}_1\right)}^4=-1.$$ The isocahedral group, $$I=\text{the isocahedral group (order 120),}$$ is generated by the matrices $$S_1=\frac{1}{\sqrt[\mathrm{}]{5}}\left(\begin{array}{cc}{\eta }^4-\eta & {\eta }^2-{\eta }^3\\ {\eta }^2-{\eta }^3& \eta -{\eta }^4\end{array}\right),T_1=\frac{1}{\sqrt[\mathrm{}]{5}}\left(\begin{array}{cc}{\eta }^2-{\eta }^4& {\eta }^4-1\\ 1-\eta & {\eta }^3-\eta \end{array}\right),S_1{T}_1=\left(\begin{array}{cc}-{\eta }^3& 0\\ 0& -{\eta }^2\end{array}\right)$$ where $\eta =e^{2\pi i/5}$. These matrices satisfy the relations $$S_1^2=-1,T_1^3=1,{\left(S_1{T}_1\right)}^5=-1.$$ It is useful to note that

1. As given, these all consist of unitary matrices (please check) so that they are subgroups of $U_2\left(ℂ\right)$. This measn that they preserve the usual hermitian inner product on $V$ and so we can take $x_1,x_2$ as an orthonormal basis of $V$.

$I\cong W{H}_3$ is a twofold cover of the alternating group $A_5$ and $$I_2\left(4\right)⊲T⊲O.$$ Apparently the generating invariants for $T$, $O$ and $I$ were given by F. Klein around 1900, I thik they can be found in the book of Orlik and Terao [OT]. Each of $T$, $O$ and $I$ have three basic invariants $$f,h=\text{Hessian of} f,t=\text{Jacobian of} f \text{and} h$$ which have degrees $$\begin{array}{ccc}\text{case} T& \text{case} O& \text{case} I\\ 4& 6& 12\\ 4& 8& 20\\ 6& 12& 30\end{array}$$ In terms of these three invariants of $T$, $O$ and $I$, we can specify the generating invariants of $G_4,\dots ,G_{22}$:
Case T: $$\begin{array}{rcc}\text{Group}& \text{Generating invariants}& \text{Degrees}\\ G_4& f,t& 4,6\\ G_5& f^3,t& 12,6\\ G_6& f,t^2& 4,12\\ G_7& f^3,t^2& 12,12\end{array}$$ Case O: $$\begin{array}{rcc}\text{Group}& \text{Generating invariants}& \text{Degrees}\\ G_8& h,t& 8,12\\ G_9& h,h^2& 8,24\\ G_{10}& h^3,t& 24,12\\ G_{11}& h^3,t^2& 24,24\\ G_{12}& f,h& 6,8\\ G_{13}& f^2,h& 12,8\\ G_{14}& f,<msup><mi>t</mi><mn>2</mn></msup>& 6,24\\ G_{15}& f^2,t^2& 12,24\end{array}$$ Case I: $$\begin{array}{rcc}\text{Group}& \text{Generating invariants}& \text{Degrees}\\ G_{16}& h,t& 20,30\\ G_{17}& h,t^2& 20,60\\ G_{18}& h^3,t& 60,30\\ G_{19}& h^3,t^2& 60,60\\ G_{20}& f,t& 12,30\\ G_{21}& f,t^2& 12,60\\ G_{22}& f,h& 12,20\end{array}$$ The groups which are exceptional real reflection groups are $$\begin{array}{cc}\text{Group}& \text{Degrees}\\ G_{23}=W{H}_3& 2,6,10\\ G_{28}=W{F}_4& 2,6,8,12\\ G_{30}=W{H}_4& 2,12,20,30\\ G_{35}=W{E}_6& 2,5,6,8,9,12\\ G_{36}=W{E}_7& 2,6,8,10,12,14,18\\ G_{37}=W{E}_8& 2,8,12,14,18,20,24,30\end{array}$$ Shephard-Todd refer to [Cox] for the invariants. Are these in [OT]? It would be good to use orthonormal bases for $V$ as in Bourbaki Chapt.4-6 (Group $G_{37}$ is the last group in the Shephard-Todd list).

The dihedral groups $G(r,r,2)$

Let $r$ be a positive integer and let $$\theta =\pi /r\text{and}\xi =e^{i2\theta }.$$ With respect to the orthonormal basis $\left\{{\epsilon }_1,{\epsilon }_2\right\}$ of ${ℂ}^2$ the dihedral group $G(r,r,2)$ is the group of $2\times 2$ matrices given by $$G\left(r,r,2\right)=\left\{\left(\begin{array}{cc}-cos2k\theta & sin2k\theta \\ sin2k\theta & cos2k\theta & \end{array}\right),\left(\begin{array}{cc}cos2k\theta & -sin2k\theta \\ sin2k\theta & cos2k\theta \end{array}\right)\mid k=0,1,\dots ,r-1\right\}$$ In this form $G(r,r,2)$ is the group of symmetries of a regular $r$-gon (embedded in ${ℝ}^2$ with its center at the origin), $$\text{Picture goes here}$$ with $s_1$ being the reflection in $H_{{\alpha }_1}$ and $s_2$ being the reflection in $H_{{\alpha }_2}$

Let $x_1$ and $x_2$ be given by $$\begin{array}{ccccc}{\epsilon }_1=\frac{1}{\sqrt[\mathrm{}]{2}}\left({\xi }^{1/2}{x}_1-{\xi }^{-1/2}{x}_2\right),& & & & x_2=\frac{{\xi }^{-1/2}}{\sqrt[\mathrm{}]{2}}\left({\epsilon }_1-i{\epsilon }_2\right),\\ & & \text{and}& & \\ {\epsilon }_2=\frac{1}{\sqrt[\mathrm{}]{2}}\left({\xi }^{1/2}{x}_1+{\xi }^{-1/2}{x}_2\right),& & & & x_2=\frac{{\xi }^{1/2}}{\sqrt[\mathrm{}]{2}}\left({\epsilon }_1+i{\epsilon }_2\right).\end{array}$$ Then, with respect to the basis $\left\{x_1,x_2\right\}$, $$G(r,r,2)=\left\{\left(\begin{array}{cc}{\xi }^k& 0\\ 0& {\xi }^{-k}\end{array}\right),\left(\begin{array}{cc}0& {\xi }^k\\ {\xi }^{-k}& 0\end{array}\right)\mid k=0,1,\dots ,r-1\right\}.$$

The roots are $${\beta }_k=cos\left(k\theta \right){\epsilon }_1+sin\left(k\theta \right){\epsilon }_2=\frac{1}{sin\left(\theta \right)}\left(sin\left(\left(k+1\right)\theta \right){\alpha }_1+sin\left(k\theta \right){\alpha }_2\right),0\le k\le 2r-1,$$ and if the positive roots are $$R^{+}=\left\{{\beta }_k\mid 0\le k\le r-1\right\}\text{then}\begin{array}{l}{\alpha }_1={\beta }_0={\epsilon }_1=\frac{1}{\sqrt[\mathrm{}]{2}}\left({\xi }^{1/2}{x}_1-{\xi }^{-1/2}{x}_2\right),\\ {\alpha }_2={\beta }_{r-1}=-cos\left(\theta \right){\epsilon }_1+sin\left(\theta \right){\epsilon }_2=\frac{1}{\sqrt[\mathrm{}]{2}}\left(x_1-x_2\right),\end{array}$$ are the simple roots with $${\beta }_k=\frac{1}{sin\left(\theta \right)}\left(sin\left(\left(k+1\right)\theta \right){\alpha }_1+sin\left(k\theta \right){\alpha }_2\right),0\le k\le 2r-1.$$ The simple reflections are $$s_1=\left(\begin{array}{cc}0& \xi \\ {\xi }^{-1}& 0\end{array}\right)\text{and}{s}_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\text{in the basis} \left\{x_1,x_2\right\},$$ and $$s_1=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\text{and}{s}_2=\left(\begin{array}{cc}-cos2\theta & sin2\theta \\ sin2\theta & cos2\theta \end{array}\right),\text{in the basis} \left\{{\epsilon }_1,{\epsilon }_2\right\}.$$ Thus ${\epsilon }_1,{\epsilon }_2$ are the eivenvectors of $s_1$ and $x_1,x_2$ are the eivenvectors of $s_1{s}_2$. Then $$-{\beta }_k={\beta }_{r+k},s_1{\beta }_k={\beta }_{r-k},s_2{\beta }_k={\beta }_{r-2-k}.$$ The elements ${\epsilon }_1,{\epsilon }_2$ satisfy $$\underset{\underset{r\text{factors}}{⏟}}{\phantom{(}{s}_1{s}_2{s}_1{s}_2\dots }=\underset{\underset{r\text{factors}}{⏟}}{\phantom{(}{s}_2{s}_1{s}_2{s}_1\dots },s_1^2=1,s_2^2=1,$$ and $t=s_1{s}_2$ and $s=s_2$ satisfy $$t^r=1,s^2=1,st=t^{-1}s.$$

The invariants are given by $$f_1=x_1^r+x_2^r=Re\left({\left({\epsilon }_1+i{\epsilon }_2\right)}^r\right),\text{and}{f}_2=x_1{x}_2=-\frac{1}{2}\left({\epsilon }_1^2+{\epsilon }_2^2\right).$$ Another choice for the invariant of degree $r$ is $${f\text{'}}_1=\prod _{i=0}^{r-1}\left(cos\left(2k\theta \right){\epsilon }_1+sin\left(2k\theta \right){\epsilon }_2\right).$$ The Cartan matrix of $I_2\left(m\right)$ is

References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[Cox] H. S. M. Coxeter, The product of the generators of a finite group generated by reflections, Duke Math. J. 18 (1951), 765–782. MR0045109 (13,528d)

[OT] P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 300. Springer-Verlag, Berlin, 1992. MR1217488 (94e:52014)

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