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 (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 ${\displaystyle r\in GL(V)}$ of finite order that fixes a complex hyperplane pointwise. I.e., the fixed-space ${\displaystyle \operatorname {Fix} (r):=\operatorname {ker} (r-\operatorname {Id} _{V})}$ has codimension 1.

A (finite) complex reflection group ${\displaystyle W\subseteq GL(V)}$ is a finite subgroup of ${\displaystyle GL(V)}$ 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 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 ${\displaystyle h}$ of an irreducible complex reflection group W of rank ${\displaystyle n}$ is defined as ${\displaystyle h={\frac {|{\mathcal {R}}|+|{\mathcal {A}}|}{n}}}$ where ${\displaystyle {\mathcal {R}}}$ denotes the set of reflections and ${\displaystyle {\mathcal {A}}}$ 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 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 (θa1a2, ...,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 ${\displaystyle S(m,n).}$

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

• G(1,1,n) is the Coxeter group An−1 = {3,3,...,3,3] = ....
• G(2,1,n) is the Coxeter group Bn = [3,3,...,3,4] = Cn = ....
• G(2,2,n) is the Coxeter group Dn = [3,3,...,31,1] = ....
• G(m,p,1) is a cyclic group of order m/p = [m/p]+.
• G(p,p,2) is the Coxeter group I2(p) = [p] = (and the Weyl group G2 when p = 6).
• G(p,p,3) is the Shephard group or .
• G(p,p,4) is the Shephard group or .
• G(p,p,n) is the Shephard group ... or ....
• 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.

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 Order Reflections Degrees Codegrees
1 n−1 Symmetric group G(1,1,n) = Sym(n) n! 2n(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 2mn(n−1)/2,dnφ(d) (d|m/pd > 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
3 1 Cyclic group G(m,1,1) = Zm m dφ(d) (d|md > 1) m 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 〈3,3,2〉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 192 218412 8,24 0,16
10 2 Z12.O = 4[4]3 or 288 26316412 12,24 0,12
11 2 Z24.O = 〈4,3,2〉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 = 〈4,3,2〉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 = 〈4,3,2〉6 288 218316 12,24 0,24
16 2 Z10.I = 5[3]5 or , <2,3,5>×Z5. 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 = 〈5,3,2〉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 = 〈5,3,2〉2 240 230 12,20 0,28
23 3 W(H3) = Z2 × PSL2(5),
Coxeter [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)
Weyl [3,4,3],
1152 212+12 2,6,8,12 0,4,6,10
29 4 W(N4) = (Z4*21 + 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
Coxeter [5,3,3],
14400 260 2,12,20,30 0,10,18,28
31 4 W(EN4) = W(O4) = (Z4*21 + 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
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,
Weyl [32,2,1],
51840 236 2,5,6,8,9,12 0,3,4,6,7,10
36 7 W(E7) = Z2 ×Sp6(2),
Weyl [33,2,1],
2903040 263 2,6,8,10,12,14,18 0,4,6,8,10,12,16
37 8 W(E8)= Z2.O+
8
(2),
Weyl [34,2,1],
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 ${\displaystyle \ell }$ being the rank of the reflection group, the degrees ${\displaystyle d_{1}\leq d_{2}\leq \ldots \leq d_{\ell }}$ 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 di satisfy the formula ${\displaystyle \prod _{i=1}^{\ell }(q+d_{i}-1)=\sum _{w\in W}q^{\dim(V^{w})}.}$

Codegrees

For ${\displaystyle \ell }$ being the rank of the reflection group, the codegrees ${\displaystyle d_{1}^{*}\geq d_{2}^{*}\geq \ldots \geq d_{\ell }^{*}}$ of W can be defined by ${\displaystyle \prod _{i=1}^{\ell }(q-d_{i}^{*}-1)=\sum _{w\in W}\det(w)q^{\dim(V^{w})}.}$

• 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 ${\displaystyle d_{i}+d_{i}^{*}=d_{\ell }}$ for all ${\displaystyle 1\leq i\leq \ell }$. 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, ${\displaystyle h=d_{\ell }}$. 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 p1, …, pn and q1, …, qn − 1 such that there is a generating set s1, …, sn satisfying the relations

${\displaystyle (s_{i})^{p_{i}}=1}$ for i = 1, …, n,
${\displaystyle s_{i}s_{j}=s_{j}s_{i}}$ if ${\displaystyle |i-j|>1}$,

and

${\displaystyle s_{i}s_{i+1}s_{i}s_{i+1}\cdots =s_{i+1}s_{i}s_{i+1}s_{i}\cdots }$ where the products on both sides have qi terms, for i = 1, …, n − 1.

This information is sometimes collected in the Coxeter-type symbol p1[q1]p2[q2] … [qn − 1]pn, 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.[1][2]

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.[3]

For example, the rank 1 group, p[], , is defined by a 1×1 matrix [1-${\displaystyle e^{2\pi i/p}}$].

Given: ${\displaystyle \zeta _{p}=e^{2\pi i/p},\omega \ \zeta _{3}=e^{2\pi i/3}={\tfrac {1}{2}}(-1+i{\sqrt {3}}),\zeta _{4}=e^{2\pi i/4}=i,\zeta _{5}=e^{2\pi i/5}={\tfrac {1}{4}}(\left({\sqrt {5}}-1\right)+i{\sqrt {2(5+{\sqrt {5}})}}),\tau ={\tfrac {1+{\sqrt {5}}}{2}},\lambda ={\tfrac {1+i{\sqrt {7}}}{2}},\omega ={\tfrac {-1+i{\sqrt {3}}}{2}}}$.

Rank 1
Group Cartan Group Cartan
2[] ${\displaystyle \left[{\begin{matrix}2\end{matrix}}\right]}$ 3[] ${\displaystyle \left[{\begin{matrix}1-\omega \end{matrix}}\right]}$
4[] ${\displaystyle \left[{\begin{matrix}1-i\end{matrix}}\right]}$ 5[] ${\displaystyle \left[{\begin{matrix}1-\zeta _{5}\end{matrix}}\right]}$
Rank 2
Group Cartan Group Cartan
G4 3[3]3 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &1\\-\omega &1-\omega \end{smallmatrix}}\right]}$ G5 3[4]3 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &1\\-2\omega &1-\omega \end{smallmatrix}}\right]}$
G6 2[6]3 ${\displaystyle \left[{\begin{smallmatrix}2&1\\1-\omega +i\omega ^{2}&1-\omega \end{smallmatrix}}\right]}$ G8 4[3]4 ${\displaystyle \left[{\begin{smallmatrix}1-i&1\\-i&1-i\end{smallmatrix}}\right]}$
G9 2[6]4 ${\displaystyle \left[{\begin{smallmatrix}2&1\\(1+{\sqrt {2}})\zeta _{8}&1+i\end{smallmatrix}}\right]}$ G10 3[4]4 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &1\\-i-\omega &1-i\end{smallmatrix}}\right]}$
G14 3[8]2 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &1\\1-\omega +\omega ^{2}{\sqrt {2}}&2\end{smallmatrix}}\right]}$ G16 5[3]5 ${\displaystyle \left[{\begin{smallmatrix}1-\zeta _{5}&1\\-\zeta _{5}&1-\zeta _{5}\end{smallmatrix}}\right]}$
G17 2[6]5 ${\displaystyle \left[{\begin{smallmatrix}2&1\\1-\zeta _{5}-i\zeta ^{3}&1-\zeta _{5}\end{smallmatrix}}\right]}$ G18 3[4]5 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &1\\-\omega -\zeta _{5}&1-\zeta _{5}\end{smallmatrix}}\right]}$
G20 3[5]3 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &1\\\omega (\tau -2)&1-\omega \end{smallmatrix}}\right]}$ G21 2[10]3 ${\displaystyle \left[{\begin{smallmatrix}2&1\\1-\omega -i\omega ^{2}\tau &1-\omega \end{smallmatrix}}\right]}$
Rank 3
Group Cartan Group Cartan
G22 <5,3,2>2 ${\displaystyle \left[{\begin{smallmatrix}2&\tau +i-1&-i+1\\-\tau -i-1&2&i\\i-1&-i&2\end{smallmatrix}}\right]}$ G23 [5,3] ${\displaystyle \left[{\begin{smallmatrix}2&-\tau &0\\-\tau &2&-1\\0&-1&2\end{smallmatrix}}\right]}$
G24 [1 1 14]4 ${\displaystyle \left[{\begin{smallmatrix}2&-1&-\lambda \\-1&2&-1\\1+\lambda &-1&2\end{smallmatrix}}\right]}$ G25 3[3]3[3]3 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &\omega ^{2}&0\\-\omega ^{2}&1-\omega &-\omega ^{2}\\0&\omega ^{2}&1-\omega \end{smallmatrix}}\right]}$
G26 3[3]3[4]2 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &-\omega ^{2}&0\\\omega ^{2}&1-\omega &-1\\0&-1+\omega &2\end{smallmatrix}}\right]}$ G27 [1 1 15]4 ${\displaystyle \left[{\begin{smallmatrix}2&-\tau &-\omega \\-\tau &2&-\omega ^{2}\\-\omega ^{2}&\omega &2\end{smallmatrix}}\right]}$
Rank 4
Group Cartan Group Cartan
G28 [3,4,3] ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}}\right]}$ G29 [1 1 2]4 ${\displaystyle \left[{\begin{smallmatrix}2&-1&i+1&0\\-1&2&-i&0\\-i+1&i&2&-1\\0&0&-1&2\end{smallmatrix}}\right]}$
G30 [5,3,3] ${\displaystyle \left[{\begin{smallmatrix}2&-\tau &0&0\\-\tau &2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}}\right]}$ G32 3[3]3[3]3 ${\displaystyle \left[{\begin{smallmatrix}1-\omega &\omega ^{2}&0&0\\-\omega ^{2}&1-\omega &-\omega ^{2}&0\\0&\omega ^{2}&1-\omega &\omega ^{2}\\0&0&-\omega ^{2}&1-\omega \end{smallmatrix}}\right]}$
Rank 5
Group Cartan Group Cartan
G31 O4 ${\displaystyle \left[{\begin{smallmatrix}2&-1&i+1&0&-i+1\\-1&2&-i&0&0\\-i+1&i&2&-1&-i+1\\0&0&-1&2&-1\\i+1&0&i+1&-1&2\end{smallmatrix}}\right]}$ G33 [1 2 2]3 ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&-1&0\\0&-1&2&-\omega &0\\0&-1&-\omega ^{2}&2&-\omega ^{2}\\0&0&0&-\omega &2\end{smallmatrix}}\right]}$

References

1. ^ 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]
2. ^ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, 1974.
3. ^ Unitary Reflection Groups, pp.91-93