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 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.

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 of an irreducible complex reflection group W of rank is defined as where denotes the set of reflections and 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.[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. A. Todd (1954). 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.[2] 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.[3]

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.

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:[4]

  • G(1, 1, n) has type An1 = [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).

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(2p, 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.

  • ST is the ShephardTodd 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 names Order Reflections Degrees Codegrees
1 n1 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(n1)/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) = Zpp[] or p dφ(d) (d|p, d > 1) p 0
4 2 W(L2), Z2.T3[3]3 or , ⟨2,3,3⟩ 24 38 4,6 0,2
5 2 Z6.T3[4]3 or 72 316 6,12 0,6
6 2 Z4.T3[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.O4[3]4 or 96 26412 8,12 0,4
9 2 Z8.O4[6]2 or or ⟨2,3,4⟩4 192 218412 8,24 0,16
10 2 Z12.O4[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.O3[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⟩×Z55[3]5 or 600 548 20,30 0,10
17 2 Z20.I5[6]2 or 1200 230548 20,60 0,40
18 2 Z30.I5[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.I3[5]3 or 360 340 12,30 0,18
21 2 Z12.I3[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) Hessian3[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
6
(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
6
(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+
8
(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 (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).

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 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 di 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

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.

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 p1, ..., pn and q1, ..., qn − 1 such that there is a generating set s1, ..., sn satisfying the relations

for i = 1, ..., n,
if ,

and

where the products on both sides have qi terms, for i = 1, ..., n − 1.

This information is sometimes collected in the Coxeter-type symbol p1[q1]p2[q2] ... [qn − 1]pn, as seen in the table above.

Among groups in the infinite family G(m, p, n), the Shephard groups are those in which p = 1. There are also 18 exceptional Shephard groups, of which three are real.[5][6]

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.[7] For example, the rank 1 group of order p (with symbols p[], ) is defined by the 1 × 1 matrix .

Given: .

Rank 1
GroupCartanGroupCartan
2[] 3[]
4[] 5[]
Rank 2
GroupCartanGroupCartan
G4 3[3]3 G5 3[4]3
G6 2[6]3 G8 4[3]4
G9 2[6]4 G10 3[4]4
G14 3[8]2 G16 5[3]5
G17 2[6]5 G18 3[4]5
G20 3[5]3 G21 2[10]3
Rank 3
GroupCartanGroupCartan
G22 <5,3,2>2 G23 [5,3]
G24 [1 1 14]4 G25 3[3]3[3]3
G26 3[3]3[4]2 G27 [1 1 15]4
Rank 4
GroupCartanGroupCartan
G28 [3,4,3] G29 [1 1 2]4
G30 [5,3,3] G32 3[3]3[3]3
Rank 5
GroupCartanGroupCartan
G31 O4 G33 [1 2 2]3

See also

References

  1. Lehrer and Taylor, Theorem 1.27.
  2. Lehrer and Taylor, p. 271.
  3. Lehrer and Taylor, Section 2.2.
  4. Lehrer and Taylor, Example 2.11.
  5. Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische Annalen. March 2002, Volume 322, Issue 3, pp 477–492. DOI:10.1007/s002080200001
  6. Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, 1974.
  7. Unitary Reflection Groups, pp.91-93
  • Broué, Michel; Malle, Gunter; Rouquier, Raphaël (1995), "On complex reflection groups and their associated braid groups" (PDF), Representations of groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Providence, R.I.: American Mathematical Society, pp. 1–13, MR 1357192
  • Broué, Michel; Malle, Gunter; Rouquier, Raphaël (1998), "Complex reflection groups, braid groups, Hecke algebras", Journal für die reine und angewandte Mathematik, 1998 (500): 127–190, CiteSeerX 10.1.1.128.2907, doi:10.1515/crll.1998.064, ISSN 0075-4102, MR 1637497
  • Deligne, Pierre (1972), "Les immeubles des groupes de tresses généralisés", Inventiones Mathematicae, 17 (4): 273–302, Bibcode:1972InMat..17..273D, doi:10.1007/BF01406236, ISSN 0020-9910, MR 0422673, S2CID 123680847
  • Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN 0-273-08517-4*
  • Lehrer, Gustav I.; Taylor, Donald E. (2009), Unitary reflection groups, Australian Mathematical Society Lecture Series, vol. 20, Cambridge University Press, ISBN 978-0-521-74989-3, MR 2542964
  • Shephard, G. C.; Todd, J. A. (1954), "Finite unitary reflection groups", Canadian Journal of Mathematics, Canadian Mathematical Society, 6: 274–304, doi:10.4153/CJM-1954-028-3, ISSN 0008-414X, MR 0059914, S2CID 3342221
  • Coxeter, Finite Groups Generated by Unitary Reflections, 1966, 4. The Graphical Notation, Table of n-dimensional groups generated by n Unitary Reflections. pp. 422–423
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