In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a complex reflection group is a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 acting on a finite-dimensional complex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

 in space pointwise. (Complex reflections are sometimes called pseudo reflections or unitary reflections or sometimes just reflections.)

Classification

Any real reflection group becomes a complex reflection group if we extend the scalars from
R to C. In particular all Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

s or Weyl group

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

s give examples of complex reflection groups.

Any finite 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 finite irreducible complex reflection groups were classified by . 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

Generalized symmetric group

In mathematics, the generalized symmetric group is the wreath product S := Z_m \wr S_n of the cyclic group of order m and the symmetric group on n letters.- Examples :...

 

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

• G(1,1,n) is the Coxeter group A_n−1
• G(2,1,n) is the Coxeter group B_n = C_n
• G(2,2,n) is the Coxeter group D_n
• G(m,p,1) is a cyclic group of order m/p.
• G(m,m,2) is the Coxeter group I₂(m) (and the Weyl group G₂ when m = 6).
• 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
  Central product
  In mathematics, especially in the field of group theory, the central product is way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central...
  
   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
  Extra special group
  In group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two extra special groups of order p1+2n. Extra special groups often occur in centralizers of...
  
  .
• 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 Symmetric group In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...  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/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
3	1	Cyclic group G(m,1,1) = Zm	m	dφ(d) (d|m, d > 1)	m	0
4	2	Z2.T = 3[3]3	24	38	4,6	0,2
5	2	Z6.T = 3[4]3	72	316	6,12	0,6
6	2	Z4.T = 3[6]2	48	2638	4,12	0,8
7	2	Z12.T = 〈3,3,3〉2	144	26316	12,12	0,12
8	2	Z4.O = 4[3]4	96	26412	8,12	0,4
9	2	Z8.O = 4[6]2	192	218412	8,24	0,16
10	2	Z12.O = 4[4]3	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)	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	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	600	548	20,30	0,10
17	2	Z20.I = 5[6]2	1200	230548	20,60	0,40
18	2	Z30.I = 5[4]3	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	360	340	12,30	0,18
21	2	Z12.I = 3[10]2	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	120	215	2,6,10	0,4,8
24	3	W(J3(4)) = Z2 × PSL2(7), Klein (2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group.A note on terminology – the "...	336	221	4,6,14	0,8,10
25	3	W(L3) = W(P3) = 31+2.SL2(3), Hessian	648	324	6,9,12	0,3,6
26	3	W(M3) =Z2 ×31+2.SL2(3), Hessian	1296	29 324	6,12,18	0,6,12
27	3	W(J3(5)) = Z2 ×(Z3.Alt(6)), Valentiner Valentiner group In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by in the form of an action of A6 on the complex projective plane, and was studied further by ....	2160	245	6,12,30	0,18,24
28	4	W(F4) = (SL2(3)* SL2(3)).(Z2 × Z2) Weyl	1152	212+12	2,6,8,12	0,4,6,10
29	4	W(N4) = (Z4*21 + 4).Sym(5)	7680	240	4,8,12,20	0,8,12,16
30	4	W(H4) = (SL2(5)*SL2(5)).Z2 Coxeter	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)	155520	380	12,18,24,30	0,6,12,18
33	5	W(K5) = Z2 ×Ω5(3) = Z2 × PSp4(3) = Z2 × PSU4(2)	51840	245	4,6,10,12,18	0,6,8,12,14
34	6	W(K6)= Z3.Ω(3).Z2, Mitchell's group Mitchell's group In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by . It has the structure 6.PSU4.2....	39191040	2126	6,12,18,24,30,42	0,12,18,24,30,36
35	6	W(E6) = SO5(3) = O(2) = PSp4(3).Z2 = PSU4(2).Z2, Weyl	51840	236	2,5,6,8,9,12	0,3,4,6,7,10
36	7	W(E7) = Z2 ×Sp6(2), Weyl	2903040	263	2,6,8,10,12,14,18	0,4,6,8,10,12,16
37	8	W(E8)= Z2.O(2), Weyl	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 .

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

Chevalley–Shephard–Todd theorem

In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex...

). For being the rank of the reflection group, the degrees 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

Codegrees

For being the rank of the reflection group, the codegrees of W can be defined by

• 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

An irreducible complex reflection group of rank is generated by or by reflections. It is said to be well-generated if it is generated by reflections ; it is proved that this is equivalent to the property for all . For irreducible well-generated complex reflection groups, the Coxeter number is defined to be 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. Any finite real reflection group is well-generated.

External links

• MAGMA Computational Algebra System page