Complex reflection group

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In mathematics, a complex reflection group is a group acting on a finite-dimensional complex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane in space pointwise. (Complex reflections are sometimes called pseudo reflections or unitary reflections or sometimes just reflections.)

Classification[edit]

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.

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 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 (θa1a2, ...,θan), by the symmetric group Sn acting by permutations of the coordinates, where θ is a primitive mth root of unity and Σai≡ 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 An−1
  • G(2,1,n) is the Coxeter group Bn = Cn
  • G(2,2,n) is the Coxeter group Dn
  • 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[edit]

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 names Order 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/pd > 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|md > 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 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 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
6
(3).Z2, Mitchell's group
39191040 2126 6,12,18,24,30,42 0,12,18,24,30,36
35 6 W(E6) = SO5(3) = O
6
(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+
8
(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 (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).

Degrees[edit]

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 \ell being the rank of the reflection group, the degrees d_1 \leq d_2 \leq \ldots \leq d_\ell 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 di satisfy the formula \prod_{i=1}^\ell(q+d_i-1)= \sum_{w\in W}q^{\dim(V^w)}.

Codegrees[edit]

For \ell being the rank of the reflection group, the codegrees d^*_1 \geq d^*_2 \geq \ldots \geq d^*_\ell of W can be defined by \prod_{i=1}^\ell(q-d^*_i-1)= \sum_{w\in 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[edit]

An irreducible complex reflection group of rank \ell is generated by \ell or by \ell+1 reflections. It is said to be well-generated if it is generated by \ell reflections ; it is proved that this is equivalent to the property d_i + d^*_i = d_\ell for all 1 \leq i \leq \ell. For irreducible well-generated complex reflection groups, the Coxeter number h is defined to be the largest degree, h := d_\ell. 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.

References[edit]

External links[edit]