Complex Reflection Group - Classification

Classification

Any real reflection group becomes a complex reflection group if we extend the scalars from R to C. In particular all Coxeter groups or Weyl groups give examples of complex reflection groups.

Any finite complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces. So it is sufficient to classify the irreducible complex reflection groups.

The finite irreducible complex reflection groups were classified by G. C. Shephard and J. A. Todd (1954). They found an infinite family G(m,p,n) depending on 3 positive integer parameters (with p dividing m), and 34 exceptional cases, that they numbered from 4 to 37, listed below. The group G(m,p,n), of order mnn!/p, is the semidirect product of the abelian group of order mn/p whose elements are (θa₁,θa₂, ...,θa_n), by the symmetric group S_n acting by permutations of the coordinates, where θ is a primitive mth root of unity and Σa_i≡ 0 mod p; it is an index p subgroup of the generalized symmetric group

Special cases of G(m,p,n):

• G(1,1,n) is the Coxeter group A_n−1
• G(2,1,n) is the Coxeter group B_n = C_n
• G(2,2,n) is the Coxeter group D_n
• G(m,p,1) is a cyclic group of order m/p.
• G(m,m,2) is the Coxeter group I₂(m) (and the Weyl group G₂ when m = 6).
• The group G(m,p,n) acts irreducibly on Cn except in the cases m=1, n>1 (symmetric group) and G(2,2,2) (Klein 4 group), when Cn splits as a sum of irreducible representations of dimensions 1 and n−1.
• The only cases when two groups G(m,p,n) are isomorphic as complex reflection groups are that G(ma,pa,1) is isomorphic to G(mb,pb,1) for any positive integers a,b. However there are other cases when two such groups are isomorphic as abstract groups.
• The complex reflection group G(2,2,3) is isomorphic as a complex reflection group to G(1,1,4) restricted to a 3 dimensional space.
• The complex reflection group G(3,3,2) is isomorphic as a complex reflection group to G(1,1,3) restricted to a 2 dimensional space.
• The complex reflection group G(2p,p,1) is isomorphic as a complex reflection group to G(1,1,2) restricted to a 1 dimensional space.