Reflection Group Examples

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, $I2 4 = the dihedral group of order 8.$ The tetrahedral group $T= the tetrahedral group (order 24)≅S4$ is generated by the matrices $S1= i 0 0 -i , T1= 1 2 ε ε3 ε ε7 , S1T1= 1 2 ε3 ε5 ε7 ε5$ where $i={e}^{2\pi i/4}$ and $\epsilon ={e}^{2\pi i/8}$. These matrices satisfy the relations $S12 = T13 = -1, S1 T1 3 =1.$ The octahedral group, $O= the octahedral group (order 48)≅WB3,$ is generated by the matrices $S1= 1 2 i 1 -1 -i , T1= 1 2 ε ε ε3 ε7 , S1T1= ε3 0 0 ε5$ where $i={e}^{2\pi i/4}$ and $\epsilon ={e}^{2\pi i/8}$. These matrices satisfy the relations $S12 = T13 = S1 T1 4 = -1.$ The isocahedral group, $I= the isocahedral group (order 120),$ is generated by the matrices $S1= 1 5 η4 -η η2 - η3 η2 - η3 η- η4 , T1= 1 5 η2 - η4 η4 -1 1-η η3 -η , S1T1= -η3 0 0 -η2$ where $\eta ={e}^{2\pi i/5}$. These matrices satisfy the relations $S12 = -1, T13 = 1, S1 T1 5 = -1.$ 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 $I24 ⊲ T ⊲ O.$ 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 which have degrees 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: $Group Generating invariants Degrees G4 f,t 4,6 G5 f3,t 12,6 G6 f,t2 4,12 G7 f3,t2 12,12$ Case O: $Group Generating invariants Degrees G8 h,t 8,12 G9 h,h2 8,24 G10 h3,t 24,12 G11 h3,t2 24,24 G12 f,h 6,8 G13 f2,h 12,8 G14 f,t2 6,24 G15 f2,t2 12,24$ Case I: $Group Generating invariants Degrees G16 h,t 20,30 G17 h,t2 20,60 G18 h3,t 60,30 G19 h3,t2 60,60 G20 f,t 12,30 G21 f,t2 12,60 G22 f,h 12,20$ The groups which are exceptional real reflection groups are $Group Degrees G23 = WH3 2,6,10 G28 = WF4 2,6,8,12 G30 = WH4 2,12,20,30 G35 = WE6 2,5,6,8,9,12 G36 = WE7 2,6,8,10,12,14,18 G37 = WE8 2,8,12,14,18,20,24,30$ 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 $G\left(r,r,2\right)$

Let $r$ be a positive integer and let $θ=π/r and ξ=ei2θ.$ With respect to the orthonormal basis $\left\{{\epsilon }_{1},{\epsilon }_{2}\right\}$ of ${ℂ}^{2}$ the dihedral group $G\left(r,r,2\right)$ is the group of $2×2$ matrices given by $G r,r,2 = -cos2kθ sin2kθ sin2kθ cos2kθ , cos2kθ -sin2kθ sin2kθ cos2kθ ∣ k=0,1,…,r-1$ In this form $G\left(r,r,2\right)$ is the group of symmetries of a regular $r$-gon (embedded in ${ℝ}^{2}$ with its center at the origin), $Picture goes here$ 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 $ε1 = 1 2 ξ1/2 x1 - ξ-1/2 x2 , x2 = ξ-1/2 2 ε1 -iε2 , and ε2 = 1 2 ξ1/2 x1+ ξ-1/2 x2 , x2 = ξ1/2 2 ε1 +i ε2 .$ Then, with respect to the basis $\left\{{x}_{1},{x}_{2}\right\}$, $G(r,r,2) = ξk 0 0 ξ-k , 0 ξk ξ-k 0 ∣ k=0,1,…,r-1 .$

The roots are $βk= coskθ ε1 + sinkθ ε2 = 1 sinθ sink+1θ α1+ sinkθ α2 , 0≤k≤2r-1,$ and if the positive roots are $R+= βk ∣ 0≤k≤r-1 then α1= β0 = ε1= 1 2 ξ1/2 x1- ξ-1/2 x2 , α2 = βr-1 = -cosθε1 +sinθε2 = 1 2 x1-x2 ,$ are the simple roots with $βk= 1 sinθ sink+1θ α1+ sinkθ α2 , 0≤k≤2r-1.$ The simple reflections are and 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 $-βk = βr+k, s1βk = βr-k, s2βk = βr-2-k.$ The elements ${\epsilon }_{1},{\epsilon }_{2}$ satisfy and $t={s}_{1}{s}_{2}$ and $s={s}_{2}$ satisfy $tr=1, s2=1, st=t-1s.$

The invariants are given by $f1= x1r + x2r = Re ε1+iε2r , and f2= x1x2 = - 1 2 ε12 + ε22 .$ Another choice for the invariant of degree $r$ is $f'1 = ∏ i=0 r-1 cos2kθε1 + sin2kθε2 .$ The Cartan matrix of ${I}_{2}\left(m\right)$ is $A= 2 -2cosπ/m -2cosπ/m 2 and A-1= 1 2sin2π/m 1 cosπ/m cosπ/m 1 .$

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)