|
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, ...,θan), by the symmetric group Sn acting by permutations of the coordinates, where θ is a primitive mth root of unity and Σai≡ 0 mod p; it is an index p subgroup of the generalized symmetric group S(m,n). Special cases of G(m,p,n): G(1,1,n) is the Coxeter group An−1 G(2,1,n) is the Coxeter group Bn = Cn G(2,2,n) is the Coxeter group Dn 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. |
About us|Jobs|Help|Disclaimer|Advertising services|Contact us|Sign in|Website map|Search|
GMT+8, 2015-9-11 22:04 , Processed in 0.131396 second(s), 16 queries .
