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 formula_1 of finite order that fixes a complex hyperplane pointwise, that is, the "fixed-space" formula_2 has codimension 1.
A (finite) complex reflection group formula_3 is a finite subgroup of formula_4 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 formula_5 of an irreducible complex reflection group "W" of rank formula_6 is defined as formula_7 where formula_8 denotes the set of reflections and formula_9 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 . 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. 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 "m"th 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 "m""n""n"!/"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 "m"th root). Algebraically, "G"("m", "p", "n") is a semidirect product of an abelian group of order "m""n"/"p" by the symmetric group Sym("n"); the elements of the abelian group are of the form ("θ""a"1, "θ""a"2, ..., "θ""a""n"), where "θ" is a primitive "m"th root of unity and ∑"a""i" ≡ 0 mod "p", and Sym("n") acts by permutations of the coordinates.
The group "G"("m","p","n") acts irreducibly on C"n" except in the cases "m" = 1, "n" > 1 (the symmetric group) and "G"(2, 2, 2) (the Klein four-group). In these cases, C"n" 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:
In addition, when "m" = "p" and "n" = 2, the group "G"("p", "p", 2) is the dihedral group of order 2"p"; as a Coxeter group, type "I""2"("p") = ["p"] = (and the Weyl group "G"2 when "p" = 6).
Other special cases and coincidences.
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" (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"(2"p", "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.
For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in .
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 formula_10 being the "rank" of the reflection group, the degrees formula_11 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:
Codegrees.
For formula_10 being the "rank" of the reflection group, the codegrees formula_14 of W can be defined by
formula_15
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 has a minimal generating set consisting of either or reflections. In the former case, the group is said to be "well-generated".
The property of being well-generated is equivalent to the condition formula_16 for all formula_17. Thus, for example, one can read off from the classification that the group is well-generated if and only if "p" = 1 or "m".
For irreducible well-generated complex reflection groups, the "Coxeter number" defined above equals the largest degree, formula_18. 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 and such that there is a generating set satisfying the relations
and
This information is sometimes collected in the Coxeter-type symbol , as seen in the table above.
Among groups in the infinite family , the Shephard groups are those in which . 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 of order "p" (with symbols p[], ) is defined by the matrix formula_23.
Given: formula_24.