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.
of finite order that fixes a complex hyperplane pointwise, that is, the fixed-space
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
In the case of real reflection groups, this definition reduces to the usual definition of the Coxeter number for finite Coxeter systems.
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.
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.
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 (θa1, θa2, ..., θan), where θ is a primitive mth root of unity and Σai ≡ 0 mod p, and Sym(n) acts by permutations of the coordinates.
In these cases, Cn splits as a sum of irreducible representations of dimensions 1 and n − 1.
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.
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).
There are a few duplicates in the first 3 lines of this list; see the previous section for details.
For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).
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).
They also showed that many other invariants of the group are determined by the degrees as follows: For
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.
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,
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.
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 and This information is sometimes collected in the Coxeter-type symbol p1[q1]p2[q2] ... [qn − 1]pn, as seen in the table above.
[5][6] An extended Cartan matrix defines the unitary group.
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 × 1 matrix