Complex reflection group

In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.
Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups.

Definition

A reflection r of a finite-dimensional complex vector space V is an element of finite order that fixes a complex hyperplane pointwise, that is, the fixed-space has codimension 1.
A complex reflection group is a finite subgroup of that is generated by reflections.

Properties

Any real reflection group becomes a complex reflection group if we extend the scalars from
R to C. In particular, all finite Coxeter groups or Weyl groups give examples of complex reflection groups.
A complex reflection group W is irreducible if the only W-invariant proper subspace of the corresponding vector space is the origin. In this case, the dimension of the vector space is called the rank of W.
The Coxeter number of an irreducible complex reflection group W of rank is defined as where denotes the set of reflections and denotes the set of reflecting hyperplanes.
In the case of real reflection groups, this definition reduces to the usual definition of the Coxeter number for finite Coxeter systems.

Classification

Any 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 irreducible complex reflection groups were classified by Geoffrey Colin Shephard and J. A. Todd. They proved that every irreducible belonged to an infinite family G depending on 3 positive integer parameters or was one of 34 exceptional cases, which they numbered from 4 to 37. The group G is the generalized symmetric group; equivalently, it is the wreath product of the symmetric group Sym by a cyclic group of order m. As a matrix group, its elements may be realized as monomial matrices whose nonzero elements are mth roots of unity.
The group G is an index-p subgroup of G. G is of order mⁿn!/p. As matrices, it may be realized as the subset in which the product of the nonzero entries is an th root of unity. Algebraically, G is a semidirect product of an abelian group of order mⁿ/p by the symmetric group Sym; the elements of the abelian group are of the form, where θ is a primitive mth root of unity and Σa_i ≡ 0 mod p, and Sym acts by permutations of the coordinates.
The group G acts irreducibly on Cⁿ except in the cases m = 1, n > 1 and G. In these cases, Cⁿ splits as a sum of irreducible representations of dimensions 1 and n − 1.

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

[Coxeter group]s

When m = 2, the representation described in the previous section consists of matrices with real entries, and hence in these cases G is a finite Coxeter group. In particular:

G has type A_n−1 = =...; the symmetric group of order n!
G has type B_n = =...; the hyperoctahedral group of order 2ⁿn!
G has type D_n = =..., order 2ⁿn!/2.

In addition, when m = p and n = 2, the group G is the dihedral group of order 2p; as a Coxeter group, type I₂ = = .

Other special cases and coincidences

The only cases when two groups G are isomorphic as complex reflection groups are that G is isomorphic to G for any positive integers a, b. However, there are other cases when two such groups are isomorphic as abstract groups.
The groups G and G are isomorphic to the symmetric group Sym. The groups G and G are isomorphic to the symmetric group Sym. Both G and G are isomorphic to the dihedral group of order 8. And the groups G are cyclic of order 2, as is G.

List of irreducible complex reflection groups

There are a few duplicates in the first 3 lines of this list; see the previous section for details.

ST is the Shephard-Todd number of the reflection group.
Rank is the dimension of the complex vector space the group acts on.
Structure describes the structure of the group. The symbol * stands for a central product of two groups. For rank 2, the quotient by the center is the group of rotations of a tetrahedron, octahedron, or icosahedron, O = Sym, I = Alt, as stated in the table. For the notation 2¹⁺⁴, see extra special group.
Order is the number of elements of the group.
Reflections describes the number of reflections: 2⁶4¹² means that there are 6 reflections of order 2 and 12 of order 4.
Degrees gives the degrees of the fundamental invariants of the ring of polynomial invariants. For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6.

ST	Rank	Structure and names	Coxeter names	Order	Reflections	Degrees	Codegrees
1	n−1	Symmetric group G = Sym		n!	2^n/2	2, 3,...,n	0,1,...,n − 2
2	n	G ''m > 1, n'' > 1, p\|m		mⁿn!/p	2^mn/2,d^nφ	m,2m,..,m; mn/''p	0,m'',..., m if p < m; 0,m,...,m, m − n if p = m
2	2	G ''p > 1,	p2 or	2p''²	2^p,d^2φ	p; 2p	0,p
2	2	Dihedral group G ''p > 2	or	2p''	2^p	2,p	0,p-2
3	1	Cyclic group G = Z_p	p or	p	d^φ	p	0
4	2	W, Z₂.T	33 or, ⟨2,3,3⟩	24	3⁸	4,6	0,2
5	2	Z₆.T	33 or	72	3¹⁶	6,12	0,6
6	2	Z₄.T	32 or	48	2⁶3⁸	4,12	0,8
7	2	Z₁₂.T	‹3,3,3›₂ or ⟨2,3,3⟩₆	144	2⁶3¹⁶	12,12	0,12
8	2	Z₄.O	44 or	96	2⁶4¹²	8,12	0,4
9	2	Z₈.O	42 or or ⟨2,3,4⟩₄	192	2¹⁸4¹²	8,24	0,16
10	2	Z₁₂.O	43 or	288	2⁶3¹⁶4¹²	12,24	0,12
11	2	Z₂₄.O	⟨2,3,4⟩₁₂	576	2¹⁸3¹⁶4¹²	24,24	0,24
12	2	Z₂.O= GL₂	⟨2,3,4⟩	48	2¹²	6,8	0,10
13	2	Z₄.O	⟨2,3,4⟩₂	96	2¹⁸	8,12	0,16
14	2	Z₆.O	32 or	144	2¹²3¹⁶	6,24	0,18
15	2	Z₁₂.O	⟨2,3,4⟩₆	288	2¹⁸3¹⁶	12,24	0,24
16	2	Z₁₀.I, ⟨2,3,5⟩×Z₅	55 or	600	5⁴⁸	20,30	0,10
17	2	Z₂₀.I	52 or	1200	2³⁰5⁴⁸	20,60	0,40
18	2	Z₃₀.I	53 or	1800	3⁴⁰5⁴⁸	30,60	0,30
19	2	Z₆₀.I	⟨2,3,5⟩₃₀	3600	2³⁰3⁴⁰5⁴⁸	60,60	0,60
20	2	Z₆.I	33 or	360	3⁴⁰	12,30	0,18
21	2	Z₁₂.I	32 or	720	2³⁰3⁴⁰	12,60	0,48
22	2	Z₄.I	⟨2,3,5⟩₂	240	2³⁰	12,20	0,28
23	3	W = Z₂ × PSL₂	,	120	2¹⁵	2,6,10	0,4,8
24	3	W = Z₂ × PSL₂, Klein	⁴,	336	2²¹	4,6,14	0,8,10
25	3	W = W = 3¹⁺².SL₂ Hessian	333,	648	3²⁴	6,9,12	0,3,6
26	3	W =Z₂ ×3¹⁺².SL₂ Hessian	233,	1296	2⁹ 3²⁴	6,12,18	0,6,12
27	3	W = Z₂ ×, Valentiner	⁴, ⁵,	2160	2⁴⁵	6,12,30	0,18,24
28	4	W = * SL₂).	,	1152	2¹²⁺¹²	2,6,8,12	0,4,6,10
29	4	W =.Sym	⁴,	7680	2⁴⁰	4,8,12,20	0,8,12,16
30	4	W = *SL₂).Z₂	,	14400	2⁶⁰	2,12,20,30	0,10,18,28
31	4	W = W =.Sp₄		46080	2⁶⁰	8,12,20,24	0,12,16,28
32	4	W = Z₃ × Sp₄	3333,	155520	3⁸⁰	12,18,24,30	0,6,12,18
33	5	W = Z₂ ×Ω₅ = Z₂ × PSp₄= Z₂ × PSU₄	³,	51840	2⁴⁵	4,6,10,12,18	0,6,8,12,14
34	6	W= Z₃.Ω.Z₂, Mitchell's group	³,	39191040	2¹²⁶	6,12,18,24,30,42	0,12,18,24,30,36
35	6	W = SO₅ = O = PSp₄.Z₂ = PSU₄.Z₂	,	51840	2³⁶	2,5,6,8,9,12	0,3,4,6,7,10
36	7	W = Z₂ ×Sp₆	,	2903040	2⁶³	2,6,8,10,12,14,18	0,4,6,8,10,12,16
37	8	W= Z₂.O	,	696729600	2¹²⁰	2,8,12,14,18,20,24,30	0,6,10,12,16,18,22,28