Complex reflection group

Short description: Concept in mathematics

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 (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra).

Definition

A (complex) reflection r (sometimes also called pseudo reflection or unitary reflection) of a finite-dimensional complex vector space V is an element [math]\displaystyle{ r \in GL(V) }[/math] of finite order that fixes a complex hyperplane pointwise, that is, the fixed-space [math]\displaystyle{ \operatorname{Fix}(r) := \operatorname{ker}(r-\operatorname{Id}_V) }[/math] has codimension 1.

A (finite) complex reflection group [math]\displaystyle{ W \subseteq GL(V) }[/math] is a finite subgroup of [math]\displaystyle{ GL(V) }[/math] 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 [math]\displaystyle{ h }[/math] of an irreducible complex reflection group W of rank [math]\displaystyle{ n }[/math] is defined as [math]\displaystyle{ h = \frac{|\mathcal{R}|+|\mathcal{A}|}{n} }[/math] where [math]\displaystyle{ \mathcal{R} }[/math] denotes the set of reflections and [math]\displaystyle{ \mathcal{A} }[/math] 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.^[1] So it is sufficient to classify the irreducible complex reflection groups.

The irreducible complex reflection groups were classified by G. C. Shephard and J. A. Todd (1954). They proved that every irreducible belonged to an infinite family G(m, p, n) depending on 3 positive integer parameters (with p dividing m) or was one of 34 exceptional cases, which they numbered from 4 to 37.^[2] The group G(m, 1, n) is the generalized symmetric group; equivalently, it is the wreath product of the symmetric group Sym(n) 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(m, p, n) is an index-p subgroup of G(m, 1, n). G(m, p, n) is of order mⁿn!/p. As matrices, it may be realized as the subset in which the product of the nonzero entries is an (m/p)th root of unity (rather than just an mth root). Algebraically, G(m, p, n) is a semidirect product of an abelian group of order mⁿ/p by the symmetric group Sym(n); the elements of the abelian group are of the form (θ^a₁, θ^a₂, ..., θ^a_n), where θ is a primitive mth root of unity and Σa_i ≡ 0 mod p, and Sym(n) acts by permutations of the coordinates.^[3]

The group G(m,p,n) acts irreducibly on Cⁿ except in the cases m = 1, n > 1 (the symmetric group) and G(2, 2, 2) (the Klein four-group). 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 groups

When m = 2, the representation described in the previous section consists of matrices with real entries, and hence in these cases G(m,p,n) is a finite Coxeter group. In particular:^[4]

G(1, 1, n) has type A_n−1 = [3,3,...,3,3] = ...; the symmetric group of order n!
G(2, 1, n) has type B_n = [3,3,...,3,4] = ...; the hyperoctahedral group of order 2ⁿn!
G(2, 2, n) has type D_n = [3,3,...,3^1,1] = ..., order 2ⁿn!/2.

In addition, when m = p and n = 2, the group G(p, p, 2) is the dihedral group of order 2p; as a Coxeter group, type I₂(p) = [p] = (and the Weyl group G₂ when p = 6).

Other special cases and coincidences

The only cases when two groups G(m, p, n) are isomorphic as complex reflection groups^{[clarification needed]} are that G(ma, pa, 1) is isomorphic to G(mb, pb, 1) for any positive integers a, b (and both are isomorphic to the cyclic group of order m/p). However, there are other cases when two such groups are isomorphic as abstract groups.

The groups G(3, 3, 2) and G(1, 1, 3) are isomorphic to the symmetric group Sym(3). The groups G(2, 2, 3) and G(1, 1, 4) are isomorphic to the symmetric group Sym(4). Both G(2, 1, 2) and G(4, 4, 2) are isomorphic to the dihedral group of order 8. And the groups G(2p, p, 1) are cyclic of order 2, as is G(1, 1, 2).

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 (cyclic) center is the group of rotations of a tetrahedron, octahedron, or icosahedron (T = Alt(4), O = Sym(4), I = Alt(5), of orders 12, 24, 60), 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(1,1,n) = Sym(n)		n!	2^{n(n − 1)/2}	2, 3, ...,n	0,1,...,n − 2
2	n	G(m,p,n) m > 1, n > 1, p\|m (G(2,2,2) is reducible)		mⁿn!/p	2^mn(n−1)/2,d^nφ(d) (d\|m/p, d > 1)	m,2m,..,(n − 1)m; mn/p	0,m,..., (n − 1)m if p < m; 0,m,...,(n − 2)m, (n − 1)m − n if p = m
2	2	G(p,1,2) p > 1,	p[4]2 or	2p²	2^p,d^2φ(d) (d\|p, d > 1)	p; 2p	0,p
2	2	Dihedral group G(p,p,2) p > 2	[p] or	2p	2^p	2,p	0,p-2
3	1	Cyclic group G(p,1,1) = Z_p	p[] or	p	d^φ(d) (d\|p, d > 1)	p	0
4	2	W(L₂), Z₂.T	3[3]3 or , ⟨2,3,3⟩	24	3⁸	4,6	0,2
5	2	Z₆.T	3[4]3 or	72	3¹⁶	6,12	0,6
6	2	Z₄.T	3[6]2 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	4[3]4 or	96	2⁶4¹²	8,12	0,4
9	2	Z₈.O	4[6]2 or or ⟨2,3,4⟩₄	192	2¹⁸4¹²	8,24	0,16
10	2	Z₁₂.O	4[4]3 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(F₃)	⟨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	3[8]2 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	5[3]5 or	600	5⁴⁸	20,30	0,10
17	2	Z₂₀.I	5[6]2 or	1200	2³⁰5⁴⁸	20,60	0,40
18	2	Z₃₀.I	5[4]3 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	3[5]3 or	360	3⁴⁰	12,30	0,18
21	2	Z₁₂.I	3[10]2 or	720	2³⁰3⁴⁰	12,60	0,48
22	2	Z₄.I	⟨2,3,5⟩₂	240	2³⁰	12,20	0,28
23	3	W(H₃) = Z₂ × PSL₂(5)	[5,3],	120	2¹⁵	2,6,10	0,4,8
24	3	W(J₃(4)) = Z₂ × PSL₂(7), Klein	[1 1 1⁴]⁴,	336	2²¹	4,6,14	0,8,10
25	3	W(L₃) = W(P₃) = 3¹⁺².SL₂(3) Hessian	3[3]3[3]3,	648	3²⁴	6,9,12	0,3,6
26	3	W(M₃) =Z₂ ×3¹⁺².SL₂(3) Hessian	2[4]3[3]3,	1296	2⁹ 3²⁴	6,12,18	0,6,12
27	3	W(J₃(5)) = Z₂ ×(Z₃.Alt(6)), Valentiner	[1 1 1⁵]⁴, [1 1 1⁴]⁵,	2160	2⁴⁵	6,12,30	0,18,24
28	4	W(F₄) = (SL₂(3)* SL₂(3)).(Z₂ × Z₂)	[3,4,3],	1152	2¹²⁺¹²	2,6,8,12	0,4,6,10
29	4	W(N₄) = (Z₄*2^{1 + 4}).Sym(5)	[1 1 2]⁴,	7680	2⁴⁰	4,8,12,20	0,8,12,16
30	4	W(H₄) = (SL₂(5)*SL₂(5)).Z₂	[5,3,3],	14400	2⁶⁰	2,12,20,30	0,10,18,28
31	4	W(EN₄) = W(O₄) = (Z₄*2^1 + 4).Sp₄(2)		46080	2⁶⁰	8,12,20,24	0,12,16,28
32	4	W(L₄) = Z₃ × Sp₄(3)	3[3]3[3]3[3]3,	155520	3⁸⁰	12,18,24,30	0,6,12,18
33	5	W(K₅) = Z₂ ×Ω₅(3) = Z₂ × PSp₄(3)= Z₂ × PSU₄(2)	[1 2 2]³,	51840	2⁴⁵	4,6,10,12,18	0,6,8,12,14
34	6	W(K₆)= Z₃.Ω−6(3).Z₂, Mitchell's group	[1 2 3]³,	39191040	2¹²⁶	6,12,18,24,30,42	0,12,18,24,30,36
35	6	W(E₆) = SO₅(3) = O−6(2) = PSp₄(3).Z₂ = PSU₄(2).Z₂	[3^2,2,1],	51840	2³⁶	2,5,6,8,9,12	0,3,4,6,7,10
36	7	W(E₇) = Z₂ ×Sp₆(2)	[3^3,2,1],	2903040	2⁶³	2,6,8,10,12,14,18	0,4,6,8,10,12,16
37	8	W(E₈)= Z₂.O+8(2)	[3^4,2,1],	696729600	2¹²⁰	2,8,12,14,18,20,24,30	0,6,10,12,16,18,22,28

For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).

Degrees

Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (Chevalley–Shephard–Todd theorem). For [math]\displaystyle{ \ell }[/math] being the rank of the reflection group, the degrees [math]\displaystyle{ d_1 \leq d_2 \leq \ldots \leq d_\ell }[/math] of the generators of the ring of invariants are called degrees of W and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows:

The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees.
The order of a complex reflection group is the product of its degrees.
The number of reflections is the sum of the degrees minus the rank.
An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2.
The degrees d_i satisfy the formula [math]\displaystyle{ \prod_{i=1}^\ell(q+d_i-1)= \sum_{w\in W}q^{\dim(V^w)}. }[/math]

Codegrees

For [math]\displaystyle{ \ell }[/math] being the rank of the reflection group, the codegrees [math]\displaystyle{ d^*_1 \geq d^*_2 \geq \ldots \geq d^*_\ell }[/math] of W can be defined by [math]\displaystyle{ \prod_{i=1}^\ell(q-d^*_i-1)= \sum_{w\in W}\det(w)q^{\dim(V^w)}. }[/math]

For a real reflection group, the codegrees are the degrees minus 2.
The number of reflection hyperplanes is the sum of the codegrees plus the rank.

Well-generated complex reflection groups

By definition, every complex reflection group is generated by its reflections. The set of reflections is not a minimal generating set, however, and every irreducible complex reflection groups of rank $n$ has a minimal generating set consisting of either $n$ or $n + 1$ reflections. In the former case, the group is said to be well-generated.

The property of being well-generated is equivalent to the condition [math]\displaystyle{ d_i + d^*_i = d_\ell }[/math] for all [math]\displaystyle{ 1 \leq i \leq \ell }[/math]. Thus, for example, one can read off from the classification that the group $G (m, p, n)$ is well-generated if and only if p = 1 or m.

For irreducible well-generated complex reflection groups, the Coxeter number $h$ defined above equals the largest degree, [math]\displaystyle{ h = d_\ell }[/math]. A reducible complex reflection group is said to be well-generated if it is a product of irreducible well-generated complex reflection groups. Every finite real reflection group is well-generated.

Shephard groups

The well-generated complex reflection groups include a subset called the Shephard groups. These groups are the symmetry groups of regular complex polytopes. In particular, they include the symmetry groups of regular real polyhedra. The Shephard groups may be characterized as the complex reflection groups that admit a "Coxeter-like" presentation with a linear diagram. That is, a Shephard group has associated positive integers $p 1, ..., p n$ and $q 1, ..., q n - 1$ such that there is a generating set $s 1, ..., s n$ satisfying the relations

[math]\displaystyle{ (s_i)^{p_i} = 1 }[/math] for

i = 1, ..., n

,

[math]\displaystyle{ s_i s_j = s_j s_i }[/math] if [math]\displaystyle{ |i - j| \gt 1 }[/math],

and

[math]\displaystyle{ s_i s_{i + 1}s_i s_{i + 1} \cdots = s_{i + 1}s_i s_{i + 1}s_i \cdots }[/math] where the products on both sides have

q i

terms, for

i = 1, ..., n - 1

.

This information is sometimes collected in the Coxeter-type symbol $p 1 [q 1] p 2 [q 2] ... [q n - 1] p n$ , as seen in the table above.

Among groups in the infinite family $G (m, p, n)$ , the Shephard groups are those in which $p = 1$ . There are also 18 exceptional Shephard groups, of which three are real.^[5]^[6]

Cartan matrices

An extended Cartan matrix defines the unitary group. Shephard groups of rank n group have n generators. Ordinary Cartan matrices have diagonal elements 2, while unitary reflections do not have this restriction.^[7] For example, the rank 1 group of order p (with symbols p[], ) is defined by the $1 \times 1$ matrix [math]\displaystyle{ \left[1-e^{2\pi i/p}\right] }[/math].

Given: [math]\displaystyle{ \zeta_p = e^{2\pi i/p}, \omega = \zeta_3 = e^{2\pi i/3} = \tfrac{1}{2}(-1+i\sqrt{3}), \zeta_4 = e^{2\pi i/4} = i, \zeta_5 = e^{2\pi i/5} = \tfrac{1}{4}(\left(\sqrt5-1\right) + i\sqrt{2(5+\sqrt5)}), \tau = \tfrac{1+\sqrt5}{2}, \lambda = \tfrac{1+i\sqrt7}{2}, \omega = \tfrac{-1+i\sqrt3}{2} }[/math].

Rank 1
Group		Cartan	Group		Cartan
2[]		[math]\displaystyle{ \left [\begin{matrix}2\end{matrix} \right ] }[/math]	3[]		[math]\displaystyle{ \left [\begin{matrix}1-\omega\end{matrix}\right ] }[/math]
4[]		[math]\displaystyle{ \left [\begin{matrix}1-i\end{matrix}\right ] }[/math]	5[]		[math]\displaystyle{ \left [\begin{matrix}1-\zeta_5\end{matrix}\right ] }[/math]

Rank 2
Group		Cartan	Group		Cartan
G₄	3[3]3	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&1\\-\omega&1-\omega\end{smallmatrix}\right ] }[/math]	G₅	3[4]3	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&1\\-2\omega&1-\omega\end{smallmatrix}\right ] }[/math]
G₆	2[6]3	[math]\displaystyle{ \left [\begin{smallmatrix}2&1\\1-\omega+i\omega^2&1-\omega\end{smallmatrix}\right ] }[/math]	G₈	4[3]4	[math]\displaystyle{ \left [\begin{smallmatrix}1-i&1\\-i&1-i\end{smallmatrix}\right ] }[/math]
G₉	2[6]4	[math]\displaystyle{ \left [\begin{smallmatrix}2&1\\(1+\sqrt2)\zeta_8&1+i\end{smallmatrix}\right ] }[/math]	G₁₀	3[4]4	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&1\\-i-\omega&1-i\end{smallmatrix}\right ] }[/math]
G₁₄	3[8]2	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&1\\1-\omega+\omega^2\sqrt2&2\end{smallmatrix}\right ] }[/math]	G₁₆	5[3]5	[math]\displaystyle{ \left [\begin{smallmatrix}1-\zeta_5&1\\-\zeta_5&1-\zeta_5\end{smallmatrix}\right ] }[/math]
G₁₇	2[6]5	[math]\displaystyle{ \left [\begin{smallmatrix}2&1\\1-\zeta_5-i\zeta^3&1-\zeta_5\end{smallmatrix}\right ] }[/math]	G₁₈	3[4]5	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&1\\-\omega-\zeta_5&1-\zeta_5\end{smallmatrix}\right ] }[/math]
G₂₀	3[5]3	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&1\\\omega(\tau-2)&1-\omega\end{smallmatrix}\right ] }[/math]	G₂₁	2[10]3	[math]\displaystyle{ \left [\begin{smallmatrix}2&1\\1-\omega-i\omega^2\tau&1-\omega\end{smallmatrix}\right ] }[/math]

Rank 3
Group		Cartan	Group		Cartan
G₂₂	<5,3,2>₂	[math]\displaystyle{ \left [\begin{smallmatrix}2&\tau+i-1&-i+1\\-\tau-i-1&2&i\\i-1&-i&2\end{smallmatrix}\right ] }[/math]	G₂₃	[5,3]	[math]\displaystyle{ \left [\begin{smallmatrix}2&-\tau&0\\-\tau&2&-1\\0&-1&2\end{smallmatrix}\right ] }[/math]
G₂₄	[1 1 1⁴]⁴	[math]\displaystyle{ \left [\begin{smallmatrix}2&-1&-\lambda\\-1&2&-1\\1+\lambda&-1&2\end{smallmatrix}\right ] }[/math]	G₂₅	3[3]3[3]3	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&\omega^2&0\\-\omega^2&1-\omega&-\omega^2\\0&\omega^2&1-\omega\end{smallmatrix}\right ] }[/math]
G₂₆	3[3]3[4]2	[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&-\omega^2&0\\\omega^2&1-\omega&-1\\0&-1+\omega&2\end{smallmatrix}\right ] }[/math]	G₂₇	[1 1 1⁵]⁴	[math]\displaystyle{ \left [\begin{smallmatrix}2&-\tau&-\omega\\-\tau&2&-\omega^2\\-\omega^2&\omega&2\end{smallmatrix}\right ] }[/math]

Rank 4
Group			Cartan	Group			Cartan
G₂₈	[3,4,3]		[math]\displaystyle{ \left [\begin{smallmatrix}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}\right ] }[/math]	G₂₉	[1 1 2]⁴		[math]\displaystyle{ \left [\begin{smallmatrix}2&-1&i+1&0\\-1&2&-i&0\\-i+1&i&2&-1\\0&0&-1&2\end{smallmatrix}\right ] }[/math]
G₃₀	[5,3,3]		[math]\displaystyle{ \left [\begin{smallmatrix}2&-\tau&0&0\\-\tau&2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}\right ] }[/math]	G₃₂	3[3]3[3]3		[math]\displaystyle{ \left [\begin{smallmatrix}1-\omega&\omega^2&0&0\\-\omega^2&1-\omega&-\omega^2&0\\0&\omega^2&1-\omega&\omega^2\\0&0&-\omega^2&1-\omega\end{smallmatrix}\right ] }[/math]

Rank 5
Group			Cartan	Group			Cartan
G₃₁	O₄		[math]\displaystyle{ \left [\begin{smallmatrix}2&-1&i+1&0&-i+1\\-1&2&-i&0&0\\-i+1&i&2&-1&-i+1\\0&0&-1&2&-1\\i+1&0&i+1&-1&2\end{smallmatrix}\right ] }[/math]	G₃₃	[1 2 2]³		[math]\displaystyle{ \left [\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&-1&0\\0&-1&2&-\omega&0\\0&-1&-\omega^2&2&-\omega^2\\0&0&0&-\omega&2\end{smallmatrix}\right ] }[/math]

References

↑ Lehrer and Taylor, Theorem 1.27.
↑ Lehrer and Taylor, p. 271.
↑ Lehrer and Taylor, Section 2.2.
↑ Lehrer and Taylor, Example 2.11.
↑ Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische Annalen. March 2002, Volume 322, Issue 3, pp 477–492. DOI:10.1007/s002080200001 [1]
↑ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, 1974.
↑ Unitary Reflection Groups, pp.91-93

Broué, Michel; Malle, Gunter; Rouquier, Raphaël (1995), "On complex reflection groups and their associated braid groups", Representations of groups (Banff, AB, 1994), CMS Conf. Proc., 16, Providence, R.I.: American Mathematical Society, pp. 1–13, https://www.math.ucla.edu/~rouquier/papers/banff.pdf
Broué, Michel; Malle, Gunter; Rouquier, Raphaël (1998), "Complex reflection groups, braid groups, Hecke algebras", Journal für die reine und angewandte Mathematik 1998 (500): 127–190, doi:10.1515/crll.1998.064, ISSN 0075-4102
Deligne, Pierre (1972), "Les immeubles des groupes de tresses généralisés", Inventiones Mathematicae 17 (4): 273–302, doi:10.1007/BF01406236, ISSN 0020-9910, Bibcode: 1972InMat..17..273D
Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN:0-273-08517-4*
Lehrer, Gustav I.; Taylor, Donald E. (2009), Unitary reflection groups, Australian Mathematical Society Lecture Series, 20, Cambridge University Press, ISBN 978-0-521-74989-3
Shephard, G. C.; Todd, J. A. (1954), "Finite unitary reflection groups", Canadian Journal of Mathematics (Canadian Mathematical Society) 6: 274–304, doi:10.4153/CJM-1954-028-3, ISSN 0008-414X, https://books.google.com/books?id=Bi7EKLHppuYC
Coxeter, Finite Groups Generated by Unitary Reflections, 1966, 4. The Graphical Notation, Table of n-dimensional groups generated by n Unitary Reflections. pp. 422–423

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Original source: https://en.wikipedia.org/wiki/Complex reflection group. Read more

[1] Lehrer and Taylor, Theorem 1.27.

[2] Lehrer and Taylor, p. 271.

[3] Lehrer and Taylor, Section 2.2.

[4] Lehrer and Taylor, Example 2.11.

[5] Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische Annalen. March 2002, Volume 322, Issue 3, pp 477–492. DOI:10.1007/s002080200001 [1]

[6] Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, 1974.

[7] Unitary Reflection Groups, pp.91-93

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