Complex reflection group

Concept in mathematics
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).

A (complex) reflection r (sometimes also called pseudo reflection or unitary reflection) of a finite dimensional complex vector space V is an element of finite order that fixes a complex hyperplane pointwise, that is, the fixed space has codimension 1. A (finite) complex reflection group is a finite subgroup of that is generated by reflections.

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 mth 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 mnn!/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 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. The group G(m,p,n) acts irreducibly on Cn except in the cases m = 1, n > 1 (the symmetric group) and G(2, 2, 2) (the Klein four group). 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. In particular:
G(1, 1, n) has type An−1 = [3,3, ,3,3] = ; the symmetric group of order n! G(2, 1, n) has type Bn = [3,3, ,3,4] = ; the hyperoctahedral group of order 2nn! G(2, 2, n) has type Dn = [3,3, ,31,1] = , order 2nn!/2. In addition, when m = p and n = 2, the group G(p, p, 2) is the dihedral group of order 2p; as a Coxeter group, type I2(p) = [p] = (and the Weyl group G2 when p = 6).

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(2p, p, 1) are cyclic of order 2, as is G(1, 1, 2).

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 namesCoxeter namesOrder 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/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 2 2 G(p,1,2) p > 1, p[4]2 or 2p2 2p,d2φ(d) (d|p, d > 1) p; 2p 0,p 2 2 Dihedral group G(p,p,2) p > 2 [p] or 2p 2p 2,p 0,p 2 3 1 Cyclic group G(p,1,1) = Zp p[] or p dφ(d) (d|p, d > 1) p 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 ⟨2,3,3⟩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 or ⟨2,3,4⟩4 192 218412 8,24 0,16 10 2 Z12. O 4[4]3 or 288 26316412 12,24 0,12 11 2 Z24. O ⟨2,3,4⟩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 ⟨2,3,4⟩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⟨2,3,4⟩6 288 218316 12,24 0,24 16 2 Z10. I, ⟨2,3,5⟩×Z5 5[3]5 or 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 ⟨2,3,5⟩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 ⟨2,3,5⟩2 240 230 12,20 0,28 23 3 W(H3) = Z2 × PSL2(5) [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) Hessian2[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) [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 [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.Ω(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(2) = PSp4(3). Z2 = PSU4(2). Z2[32,2,1], 51840 236 2,5,6,8,9,12 0,3,4,6,7,10 36 7 W(E7) = Z2 ×Sp6(2)[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(2)[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 .

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 for all 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.