Reflection Group Examples

Reflection Group Examples

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 20 October 2010

The groups T, O and I

The rank 2 exceptional complex reflection groups G4,,G22 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=e2πi/4 and ε=e2πi/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=e2πi/4 and ε=e2πi/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 η=e2πi/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 U2. This measn that they preserve the usual hermitian inner product on V and so we can take x1,x2 as an orthonormal basis of V.
IWH3 is a twofold cover of the alternating group A5 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 f, h= Hessian of  f, t= Jacobian of f and h which have degrees case T case O case I 4 6 12 4 8 20 6 12 30 In terms of these three invariants of T, O and I, we can specify the generating invariants of G4,,G22:
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 G37 is the last group in the Shephard-Todd list).

The dihedral groups G(r,r,2)

Let r be a positive integer and let θ=π/r and ξ=ei2θ. With respect to the orthonormal basis ε1,ε2 of 2 the dihedral group G(r,r,2) 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(r,r,2) is the group of symmetries of a regular r-gon (embedded in 2 with its center at the origin), Picture goes here with s1 being the reflection in Hα1 and s2 being the reflection in Hα2

Let x1 and x2 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 x1,x2 , 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 , 0k2r-1, and if the positive roots are R+= βk 0kr-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 , 0k2r-1. The simple reflections are s1= 0 ξ ξ-1 0 and s2 = 0 1 1 0 , in the basis  x1,x2 , and s1= -1 0 0 1 and s2= -cos2θ sin2θ sin2θ cos2θ , in the basis  ε1, ε2 . Thus ε1, ε2 are the eivenvectors of s1 and x1, x2 are the eivenvectors of s1s2. Then -βk = βr+k, s1βk = βr-k, s2βk = βr-2-k. The elements ε1,ε2 satisfy ( s1 s2 s1 s2 r   factors = ( s2 s1 s2 s1 r   factors , s12 =1, s22 = 1, and t=s1s2 and s=s2 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 I2m is A= 2 -2cosπ/m -2cosπ/m 2 and A-1= 1 2sin2π/m 1 cosπ/m cosπ/m 1 .


[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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