Complex reflection group

Definition

A (complex) reflection r (sometimes also called pseudo reflection or unitary reflection) of a finite-dimensional complex vector space V is an element r ∈ G L ( V ) of finite order that fixes a complex hyperplane pointwise. I.e., the fixed-space Fix ⁡ ( r ) := ker ⁡ ( r − Id V ) has codimension 1.

A (finite) complex reflection group W ⊆ G L ( V ) is a finite subgroup of G L ( 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 h of an irreducible complex reflection group W of rank n is defined as h = | R | + | A | n where R denotes the set of reflections and 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 (θ^a₁,θ^a₂, ...,θ^a_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 S ( m , n ) .

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

G(1,1,n) is the Coxeter group A_n−1 = {3,3,...,3,3] = ....

G(2,1,n) is the Coxeter group B_n = [3,3,...,3,4] = C_n = ....

G(2,2,n) is the Coxeter group D_n = [3,3,...,3^1,1] = ....

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,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.

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 ℓ being the rank of the reflection group, the degrees d 1 ≤ d 2 ≤ … ≤ d ℓ 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 ∏ i = 1 ℓ ( q + d i − 1 ) = ∑ w ∈ W q dim ⁡ ( V w ) .

Codegrees

For ℓ being the rank of the reflection group, the codegrees d 1 ∗ ≥ d 2 ∗ ≥ … ≥ d ℓ ∗ of W can be defined by ∏ i = 1 ℓ ( q − d i ∗ − 1 ) = ∑ w ∈ 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 d i + d i ∗ = d ℓ for all 1 ≤ i ≤ ℓ . 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, h = d ℓ . 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, …, p_n and q₁, …, q_{n − 1} such that there is a generating set s₁, …, s_n satisfying the relations

( s i ) p i = 1 for i = 1, …, n, s i s j = s j s i if | i − j | > 1 ,

and

s i s i + 1 s i s i + 1 ⋯ = s i + 1 s i s i + 1 s i ⋯ 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 p₁[q₁]p₂[q₂] … [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.

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.

For example, the rank 1 group, p[], , is defined by a 1×1 matrix [1- e 2 π i / p ].

Given: ζ p = e 2 π i / p , ω   ζ 3 = e 2 π i / 3 = 1 2 ( − 1 + i 3 ) , ζ 4 = e 2 π i / 4 = i , ζ 4 = e 2 π i / 5 = 1 4 ( ( 5 − 1 ) + i 2 ( 5 + 5 ) ) , τ = 1 + 5 2 , λ = 1 + i 7 2 , ω = − 1 + i 3 2 .