Complex reflection group

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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 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.


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] = CDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png; the symmetric group of order n!
  • G(2, 1, n) has type Bn = [3,3,...,3,4] = CDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 4.pngCDel node.png; the hyperoctahedral group of order 2nn!
  • G(2, 2, n) has type Dn = [3,3,...,31,1] = CDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, 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] = CDel branch.pngCDel labelp.png (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[ clarification needed ] 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.

STRankStructure and namesCoxeter namesOrderReflectionsDegreesCodegrees
1n1 Symmetric group G(1,1,n) = Sym(n)n!2n(n  1)/22, 3, ...,n0,1,...,n  2
2nG(m,p,n) m > 1, n > 1, p|m (G(2,2,2) is reducible)mnn!/p2mn(n1)/2,dnφ(d) (d|m/p, d > 1)m,2m,..,(n  1)m; mn/p0,m,..., (n  1)m if p < m; 0,m,...,(n  2)m, (n  1)m  n if p = m
22G(p,1,2) p > 1,p[4]2 or CDel pnode.pngCDel 4.pngCDel node.png2p22p,d2φ(d) (d|p, d > 1)p; 2p0,p
22 Dihedral group G(p,p,2) p > 2[p] or CDel node.pngCDel p.pngCDel node.png2p2p2,p0,p-2
31 Cyclic group G(p,1,1) = Zpp[] or CDel pnode.pngpdφ(d) (d|p, d > 1)p0
42W(L2), Z2.T3[3]3 or CDel 3node.pngCDel 3.pngCDel 3node.png, ⟨2,3,3⟩ 24384,60,2
52Z6.T3[4]3 or CDel 3node.pngCDel 4.pngCDel 3node.png723166,120,6
62Z4.T3[6]2 or CDel 3node.pngCDel 6.pngCDel node.png4826384,120,8
72Z12.T‹3,3,3›2 or ⟨2,3,3⟩61442631612,120,12
82Z4.O4[3]4 or CDel 4node.pngCDel 3.pngCDel 4node.png96264128,120,4
92Z8.O4[6]2 or CDel 4node.pngCDel 6.pngCDel node.png or ⟨2,3,4⟩41922184128,240,16
102Z12.O4[4]3 or CDel 4node.pngCDel 4.pngCDel 3node.png2882631641212,240,12
122Z2.O= GL2(F3) ⟨2,3,4⟩ 482126,80,10
142Z6.O3[8]2 or CDel 3node.pngCDel 8.pngCDel node.png1442123166,240,18
162Z10.I, ⟨2,3,5⟩×Z55[3]5 or CDel 5node.pngCDel 3.pngCDel 5node.png60054820,300,10
172Z20.I5[6]2 or CDel 5node.pngCDel 6.pngCDel node.png120023054820,600,40
182Z30.I5[4]3 or CDel 5node.pngCDel 4.pngCDel 3node.png180034054830,600,30
202Z6.I3[5]3 or CDel 3node.pngCDel 5.pngCDel 3node.png36034012,300,18
212Z12.I3[10]2 or CDel 3node.pngCDel 10.pngCDel node.png72023034012,600,48
233W(H3) = Z2 × PSL2(5)[5,3], CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png1202152,6,100,4,8
243W(J3(4)) = Z2 × PSL2(7), Klein [1 1 14]4, CDel node.pngCDel 4split1.pngCDel branch.pngCDel label4.png3362214,6,140,8,10
253W(L3) = W(P3) = 31+2.SL2(3) Hessian 3[3]3[3]3, CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png6483246,9,120,3,6
263W(M3) =Z2 ×31+2.SL2(3) Hessian 2[4]3[3]3, CDel node.pngCDel 4.pngCDel 3node.pngCDel 3.pngCDel 3node.png129629 3246,12,180,6,12
273W(J3(5)) = Z2 ×(Z3.Alt(6)), Valentiner [1 1 15]4, CDel node.pngCDel 4split1.pngCDel branch.pngCDel label5.png
[1 1 14]5, CDel node.pngCDel 5split1.pngCDel branch.pngCDel label4.png
284W(F4) = (SL2(3)* SL2(3)).(Z2 × Z2)[3,4,3], CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png1152212+122,6,8,120,4,6,10
294W(N4) = (Z4*21 + 4).Sym(5)[1 1 2]4, CDel node.pngCDel 4split1.pngCDel branch.pngCDel 3a.pngCDel nodea.png76802404,8,12,200,8,12,16
304W(H4) = (SL2(5)*SL2(5)).Z2[5,3,3], CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png144002602,12,20,300,10,18,28
314W(EN4) = W(O4) = (Z4*21 + 4).Sp4(2)460802608,12,20,240,12,16,28
324W(L4) = Z3 × Sp4(3)3[3]3[3]3[3]3, CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png15552038012,18,24,300,6,12,18
335W(K5) = Z2 ×Ω5(3) = Z2 × PSp4(3)= Z2 × PSU4(2)[1 2 2]3, CDel node.pngCDel 3split1.pngCDel branch.pngCDel 3ab.pngCDel nodes.png518402454,6,10,12,180,6,8,12,14
346W(K6)= Z3
(3).Z2, Mitchell's group
[1 2 3]3, CDel node.pngCDel 3split1.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3a.pngCDel nodea.png3919104021266,12,18,24,30,420,12,18,24,30,36
356W(E6) = SO5(3) = O
(2) = PSp4(3).Z2 = PSU4(2).Z2
[32,2,1], CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png518402362,5,6,8,9,120,3,4,6,7,10
367W(E7) = Z2 ×Sp6(2)[33,2,1], CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png29030402632,6,8,10,12,14,180,4,6,8,10,12,16
378W(E8)= Z2.O+
[34,2,1], CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png69672960021202,8,12,14,18,20,24,300,6,10,12,16,18,22,28

For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in (MichelBroué,Gunter Malle&Raphaël Rouquier  1998 ).


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:


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

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 ,


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[], CDel pnode.png) is defined by the 1 × 1 matrix .

Given: .

Rank 1
2[]CDel node.png3[]CDel 3node.png
4[]CDel 4node.png5[]CDel 5node.png
Rank 2
G43[3]3CDel 3node.pngCDel 3.pngCDel 3node.pngG53[4]3CDel 3node.pngCDel 4.pngCDel 3node.png
G62[6]3CDel node.pngCDel 6.pngCDel 3node.pngG84[3]4CDel 4node.pngCDel 3.pngCDel 4node.png
G92[6]4CDel node.pngCDel 6.pngCDel 4node.pngG103[4]4CDel 3node.pngCDel 4.pngCDel 4node.png
G143[8]2CDel 3node.pngCDel 8.pngCDel node.pngG165[3]5CDel 5node.pngCDel 3.pngCDel 5node.png
G172[6]5CDel node.pngCDel 6.pngCDel 5node.pngG183[4]5CDel 3node.pngCDel 4.pngCDel 5node.png
G203[5]3CDel 3node.pngCDel 5.pngCDel 3node.pngG212[10]3CDel node.pngCDel 10.pngCDel 3node.png
Rank 3
G22<5,3,2>2G23[5,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
G24[1 1 14]4CDel node.pngCDel 4split1.pngCDel branch.pngCDel label4.pngG253[3]3[3]3CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
G263[3]3[4]2CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.pngG27[1 1 15]4CDel node.pngCDel 4split1.pngCDel branch.pngCDel label5.png
Rank 4
G28[3,4,3]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngG29[1 1 2]4CDel node.pngCDel 4split1.pngCDel branch.pngCDel 3a.pngCDel nodea.png
G30[5,3,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngG323[3]3[3]3CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
Rank 5
G31O4G33[1 2 2]3CDel node.pngCDel 3split1.pngCDel branch.pngCDel 3ab.pngCDel nodes.png

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  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