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.

- Definition
- Properties
- Classification
- Special cases of G(m, p, n)
- List of irreducible complex reflection groups
- Degrees
- Codegrees
- Well-generated complex reflection groups
- Shephard groups
- Cartan matrices
- References
- External links

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 *m*th 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 *m*^{n}*n*!/*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 *m*th root). Algebraically, *G*(*m*, *p*, *n*) is a semidirect product of an abelian group of order *m*^{n}/*p* by the symmetric group Sym(*n*); the elements of the abelian group are of the form (*θ*^{a1}, *θ*^{a2}, ..., *θ*^{an}), where *θ* is a primitive *m*th root of unity and Σ*a*_{i} ≡ 0 mod *p*, and Sym(*n*) acts by permutations of the coordinates.^{ [3] }

The group *G*(*m*,*p*,*n*) acts irreducibly on **C**^{n} except in the cases *m* = 1, *n* > 1 (the symmetric group) and *G*(2, 2, 2) (the Klein four-group). In these cases, **C**^{n} 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:^{ [4] }

*G*(1, 1,*n*) has type*A*_{n−1}= [3,3,...,3,3] = ...; the symmetric group of order*n*!*G*(2, 1,*n*) has type*B*_{n}= [3,3,...,3,4] = ...; the hyperoctahedral group of order 2^{n}*n*!*G*(2, 2,*n*) has type*D*_{n}= [3,3,...,3^{1,1}] = ..., order 2^{n}*n*!/2.

In addition, when *m* = *p* and *n* = 2, the group *G*(*p*, *p*, 2) is the dihedral group of order 2*p*; as a Coxeter group, type *I*_{2}(*p*) = [*p*] = (and the Weyl group *G*_{2} when *p* = 6).

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*(2*p*, *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 2^{1+4}, see extra special group.**Order**is the number of elements of the group.**Reflections**describes the number of reflections: 2^{6}4^{12}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 | Coxeter names | Order | Reflections | Degrees | Codegrees |
---|---|---|---|---|---|---|---|

1 | n−1 | Symmetric group G(1,1,n) = Sym(n) | n! | 2^{n(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) | m^{n}n!/p | 2^{mn(n−1)/2},d^{nφ(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 | 2p^{2} | 2^{p},d^{2φ(d)} (d|p, d > 1) | p; 2p | 0,p |

2 | 2 | Dihedral group G(p,p,2) p > 2 | [p] or | 2p | 2^{p} | 2,p | 0,p-2 |

3 | 1 | Cyclic group G(p,1,1) = Z_{p} | p[] or | p | d^{φ(d)} (d|p, d > 1) | p | 0 |

4 | 2 | W(L_{2}), Z_{2}.T | 3[3]3 or , ⟨2,3,3⟩ | 24 | 3^{8} | 4,6 | 0,2 |

5 | 2 | Z_{6}.T | 3[4]3 or | 72 | 3^{16} | 6,12 | 0,6 |

6 | 2 | Z_{4}.T | 3[6]2 or | 48 | 2^{6}3^{8} | 4,12 | 0,8 |

7 | 2 | Z_{12}.T | ‹3,3,3›_{2} or ⟨2,3,3⟩_{6} | 144 | 2^{6}3^{16} | 12,12 | 0,12 |

8 | 2 | Z_{4}.O | 4[3]4 or | 96 | 2^{6}4^{12} | 8,12 | 0,4 |

9 | 2 | Z_{8}.O | 4[6]2 or or ⟨2,3,4⟩_{4} | 192 | 2^{18}4^{12} | 8,24 | 0,16 |

10 | 2 | Z_{12}.O | 4[4]3 or | 288 | 2^{6}3^{16}4^{12} | 12,24 | 0,12 |

11 | 2 | Z_{24}.O | ⟨2,3,4⟩_{12} | 576 | 2^{18}3^{16}4^{12} | 24,24 | 0,24 |

12 | 2 | Z_{2}.O= GL_{2}(F_{3}) | ⟨2,3,4⟩ | 48 | 2^{12} | 6,8 | 0,10 |

13 | 2 | Z_{4}.O | ⟨2,3,4⟩_{2} | 96 | 2^{18} | 8,12 | 0,16 |

14 | 2 | Z_{6}.O | 3[8]2 or | 144 | 2^{12}3^{16} | 6,24 | 0,18 |

15 | 2 | Z_{12}.O | ⟨2,3,4⟩_{6} | 288 | 2^{18}3^{16} | 12,24 | 0,24 |

16 | 2 | Z_{10}.I, ⟨2,3,5⟩×Z_{5} | 5[3]5 or | 600 | 5^{48} | 20,30 | 0,10 |

17 | 2 | Z_{20}.I | 5[6]2 or | 1200 | 2^{30}5^{48} | 20,60 | 0,40 |

18 | 2 | Z_{30}.I | 5[4]3 or | 1800 | 3^{40}5^{48} | 30,60 | 0,30 |

19 | 2 | Z_{60}.I | ⟨2,3,5⟩_{30} | 3600 | 2^{30}3^{40}5^{48} | 60,60 | 0,60 |

20 | 2 | Z_{6}.I | 3[5]3 or | 360 | 3^{40} | 12,30 | 0,18 |

21 | 2 | Z_{12}.I | 3[10]2 or | 720 | 2^{30}3^{40} | 12,60 | 0,48 |

22 | 2 | Z_{4}.I | ⟨2,3,5⟩_{2} | 240 | 2^{30} | 12,20 | 0,28 |

23 | 3 | W(H_{3}) = Z_{2} × PSL_{2}(5) | [5,3], | 120 | 2^{15} | 2,6,10 | 0,4,8 |

24 | 3 | W(J_{3}(4)) = Z_{2} × PSL_{2}(7), Klein | [1 1 1^{4}]^{4}, | 336 | 2^{21} | 4,6,14 | 0,8,10 |

25 | 3 | W(L_{3}) = W(P_{3}) = 3^{1+2}.SL_{2}(3) Hessian | 3[3]3[3]3, | 648 | 3^{24} | 6,9,12 | 0,3,6 |

26 | 3 | W(M_{3}) =Z_{2} ×3^{1+2}.SL_{2}(3) Hessian | 2[4]3[3]3, | 1296 | 2^{9} 3^{24} | 6,12,18 | 0,6,12 |

27 | 3 | W(J_{3}(5)) = Z_{2} ×(Z_{3}.Alt(6)), Valentiner | [1 1 1^{5}]^{4}, [1 1 1 ^{4}]^{5}, | 2160 | 2^{45} | 6,12,30 | 0,18,24 |

28 | 4 | W(F_{4}) = (SL_{2}(3)* SL_{2}(3)).(Z_{2} × Z_{2}) | [3,4,3], | 1152 | 2^{12+12} | 2,6,8,12 | 0,4,6,10 |

29 | 4 | W(N_{4}) = (Z_{4}*2^{1 + 4}).Sym(5) | [1 1 2]^{4}, | 7680 | 2^{40} | 4,8,12,20 | 0,8,12,16 |

30 | 4 | W(H_{4}) = (SL_{2}(5)*SL_{2}(5)).Z_{2} | [5,3,3], | 14400 | 2^{60} | 2,12,20,30 | 0,10,18,28 |

31 | 4 | W(EN_{4}) = W(O_{4}) = (Z_{4}*2^{1 + 4}).Sp_{4}(2) | 46080 | 2^{60} | 8,12,20,24 | 0,12,16,28 | |

32 | 4 | W(L_{4}) = Z_{3} × Sp_{4}(3) | 3[3]3[3]3[3]3, | 155520 | 3^{80} | 12,18,24,30 | 0,6,12,18 |

33 | 5 | W(K_{5}) = Z_{2} ×Ω_{5}(3) = Z_{2} × PSp_{4}(3)= Z_{2} × PSU_{4}(2) | [1 2 2]^{3}, | 51840 | 2^{45} | 4,6,10,12,18 | 0,6,8,12,14 |

34 | 6 | W(K_{6})= Z_{3}.Ω^{−}_{6}(3).Z_{2}, Mitchell's group | [1 2 3]^{3}, | 39191040 | 2^{126} | 6,12,18,24,30,42 | 0,12,18,24,30,36 |

35 | 6 | W(E_{6}) = SO_{5}(3) = O^{−}_{6}(2) = PSp_{4}(3).Z_{2} = PSU_{4}(2).Z_{2} | [3^{2,2,1}], | 51840 | 2^{36} | 2,5,6,8,9,12 | 0,3,4,6,7,10 |

36 | 7 | W(E_{7}) = Z_{2} ×Sp_{6}(2) | [3^{3,2,1}], | 2903040 | 2^{63} | 2,6,8,10,12,14,18 | 0,4,6,8,10,12,16 |

37 | 8 | W(E_{8})= Z_{2}.O^{+}_{8}(2) | [3^{4,2,1}], | 696729600 | 2^{120} | 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 (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:

- 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

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.

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.

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 *p*_{1}, ..., *p*_{n} and *q*_{1}, ..., *q*_{n − 1} such that there is a generating set *s*_{1}, ..., *s*_{n} satisfying the relations

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

and

- where the products on both sides have
*q*_{i}terms, for*i*= 1, ...,*n*− 1.

This information is sometimes collected in the Coxeter-type symbol *p*_{1}[*q*_{1}]*p*_{2}[*q*_{2}] ... [*q*_{n − 1}]*p*_{n}, 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] }

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: .

Group | Cartan | Group | Cartan | ||
---|---|---|---|---|---|

2[] | 3[] | ||||

4[] | 5[] |

Group | Cartan | Group | Cartan | ||||
---|---|---|---|---|---|---|---|

G_{4} | 3[3]3 | G_{5} | 3[4]3 | ||||

G_{6} | 2[6]3 | G_{8} | 4[3]4 | ||||

G_{9} | 2[6]4 | G_{10} | 3[4]4 | ||||

G_{14} | 3[8]2 | G_{16} | 5[3]5 | ||||

G_{17} | 2[6]5 | G_{18} | 3[4]5 | ||||

G_{20} | 3[5]3 | G_{21} | 2[10]3 |

Group | Cartan | Group | Cartan | ||||
---|---|---|---|---|---|---|---|

G_{22} | <5,3,2>_{2} | G_{23} | [5,3] | ||||

G_{24} | [1 1 1^{4}]^{4} | G_{25} | 3[3]3[3]3 | ||||

G_{26} | 3[3]3[4]2 | G_{27} | [1 1 1^{5}]^{4} |

Group | Cartan | Group | Cartan | ||||
---|---|---|---|---|---|---|---|

G_{28} | [3,4,3] | G_{29} | [1 1 2]^{4} | ||||

G_{30} | [5,3,3] | G_{32} | 3[3]3[3]3 |

Group | Cartan | Group | Cartan | ||||
---|---|---|---|---|---|---|---|

G_{31} | O_{4} | G_{33} | [1 2 2]^{3} |

In abstract algebra, the **symmetric group** defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .

In mathematics, a **root system** is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

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 that 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 group. In fact it turns out that *most* finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

In mathematics, a **Coxeter group**, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

In mathematics, and especially the discipline of representation theory, the **Schur indicator**, named after Issai Schur, or **Frobenius–Schur indicator** describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has. It can be used to classify the irreducible representations of compact groups on real vector spaces.

In mathematics, the **Coxeter number***h* is the order of a **Coxeter element** of an irreducible Coxeter group. It is named after H.S.M. Coxeter.

In mathematics, **D _{3}** (sometimes alternatively denoted by

A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. Both the regular dodecahedron and the rhombic triacontahedron have the same set of symmetries.

In mathematics, a **Gelfand pair** is a pair *(G,K)* consisting of a group *G* and a subgroup *K* that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.

In geometry, a **Coxeter–Dynkin diagram** is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3, and each pair of nodes that is not connected by a branch at all represents a pair of mirrors at order-2.

In geometry, a **complex polytope** is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

In mathematics, the main results concerning irreducible **unitary representations** of the Lie group SL(2,**R**) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).

**Schur–Weyl duality** is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. It is named after two pioneers of representation theory of Lie groups, Issai Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary and general linear groups.

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 general linear group the theorem was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre.

In mathematics, a **zonal spherical function** or often just **spherical function** is a function on a locally compact group *G* with compact subgroup *K* that arises as the matrix coefficient of a *K*-invariant vector in an irreducible representation of *G*. The key examples are the matrix coefficients of the *spherical principal series*, the irreducible representations appearing in the decomposition of the unitary representation of *G* on *L*^{2}(*G*/*K*). In this case the commutant of *G* is generated by the algebra of biinvariant functions on *G* with respect to *K* acting by right convolution. It is commutative if in addition *G*/*K* is a symmetric space, for example when *G* is a connected semisimple Lie group with finite centre and *K* is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant *L*^{1} functions is larger; when *G* is a semisimple Lie group with maximal compact subgroup *K*, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

In geometry, **Coxeter notation** is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

In mathematics, the **tensor product of representations** is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.

This is a **glossary of representation theory** in mathematics.

The **affine symmetric groups** are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. Each one is an infinite extension of a finite symmetric group, the group of permutations (rearrangements) of a finite set. In addition to their geometric description, the affine symmetric groups may be defined as collections of permutations of the integers that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied as part of the fields of combinatorics and representation theory.

- ↑ Lehrer and Taylor, Theorem 1.27.
- ↑ Lehrer and Taylor, p. 271.
- ↑ Lehrer and Taylor, Section 2.2.
- ↑ Lehrer and Taylor, Example 2.11.
- ↑ 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 - ↑ Coxeter, H. S. M.;
*Regular Complex Polytopes*, Cambridge University Press, 1974. - ↑ 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

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.