CONTRIBUTION TO THE HANDBOOK OF ALGEBRA

MEINOLF GECK AND GUNTER MALLE

Chapter: Reflection groups

This chapter is concerned with the theory of finite reflection groups, that is, finite groups generated by reflections in a real or complex vector space. This is a rich theory, both for intrinsic reasons and as far as applications in other mathematical areas or mathematical physics are concerned. The origin of the theory can be traced back to the ancient study of symmetries of regular polyhedra. Another extremely important impetus comes from the theory of semisimple Lie algebras and Lie groups, where finite reflection groups occur as “Weyl groups”. In the last decade, Broué’s “abelian defect group conjecture” (a conjecture concerning the representations of finite groups over fields of positive characteristic) has lead to a vast research program, in which complex reflection groups, corresponding braid groups and Hecke algebras play a prominent role. Thus, the theory of reflection groups is at the same time a well-established classical piece of mathematics and still a very active research area. The aim of this chapter (and a subsequent one on Hecke algebras) is to give an overwiew of both these aspects.

As far as the study of reflection groups as such is concerned, there are (at least) three reasons why this leads to an interesting and rich theory:

Classification: Given a suitable notion of “irreducible” reflection groups, it is possible to give a complete classification, with typically several infinite families of groups and a certain number of exceptional cases. In fact, this classification can be seen as the simplest possible model for much more complex classification results concerning related algebraic structures, such as complex semisimple Lie algebras, simple algebraic groups and, eventually, finite simple groups. Besides the independent interest of such a classification, we mention that there is a certain number of results on finite reflection groups which can be stated in general terms but whose proof requires a case–by–case analysis according to the classification. (For example, the fact that every element in a finite real reflection group is conjugate to its inverse.)

Presentations: Reflection groups have a highly symmetric “Coxeter type presentation” with generators and defining relations (visualized by “Dynkin diagrams” or generalisations thereof), which makes it possible to study them by purely combinatorial methods (length function, reduced expressions and so on). From this point of view, the associated Hecke algebras can be seen as “deformations” of the group algebras of finite reflection groups, where one or several formal parameters are introduced into the set of defining relations. One of the most important developments in this direction is the discovery of the Kazhdan–Lusztig polynomials and the whole theory coming with them. (This is discussed in more detail in the chapter on Hecke algebras.)

Topology and geometry: The action of a reflection group on the underlying vector space opens the possibility of using geometric methods. First of all, the ring of invariant symmetric functions on that vector space always is a polynomial ring (and this characterizes finite reflection groups). Furthermore, we have a corresponding hyperplane arrangement which gives rise to the definition of an associated braid group as the fundamental group of a certain topological space. For the symmetric group, we obtain in this way the classical Artin braid group, with applications in the theory of knots and links.

Furthermore, all these aspects are related to each other which—despite being quite elementary taken individually—eventually leads to a highly sophisticated theory.

We have divided our survey into four major parts. The first part deals with finite complex or real reflection groups in general. The second part deals with finite real reflection groups and the relations with the theory of Coxeter groups. The third part is concerned with the associated braid groups. Finally, in the fourth part, we consider complex irreducible characters of finite reflection groups.

We certainly do not pretend to give a complete picture of all aspects of the theory of reflection groups and Coxeter groups. Especially, we will not say so much about areas that we do not feel competent in; at our best knowledge, we try to give at least some references for further reading in such cases. This concerns, in particular, all aspects of infinite (affine, hyperbolic …) Coxeter groups.

1. Finite groups generated by reflections

1.1. Definitions. Let V be a finite dimensional vector space over a field K. A reflection on V is a non-trivial element g ∈ GL(V) of finite order which fixes a hyperplane in V pointwise. There are two types of reflections, according to whether g is semisimple (hence diagonalisable) or unipotent. Often, the term reflection is reserved for the first type of elements, while the second are called transvections. They can only occur in positive characteristic. Here, we will almost exclusively be concerned with ground fields K of characteristic 0, which we may and will then assume to be subfields of the field ℂ of complex numbers. Then, by our definition, reflections are always semisimple and (thus) diagonalisable. Over fields K contained in the field ℝ of real numbers, reflections necessarily have order 2, which is the case motivating their name. Some authors reserve the term reflection for this case, and speak of pseudo-reflections in the case of arbitrary (finite) order.

Let g ∈ GL(V) be a reflection. The hyperplane C_{V} (g) fixed point-wise by g is called the reflecting hyperplane of g. Then V = C_{V} (g)⊕V_{g} for a unique g-invariant subspace V_{g} of V of dimension 1. Any non-zero vector v ∈ V_{g} is called a root for g. Thus, a root for a reflection is an eigenvector with eigenvalue different from 1. Now assume in addition that V is Hermitean. Then conversely, given a vector v ≠ 0 in V and a natural number n ≥ 2 we may define a reflection in V with root v and of order n by g.v := exp(2πi/n)v, and g|_{V^{⊥}} = id.

A reflection group on V is now a finite subgroup W ⩽ GL(V) generated by reflections. Note that any finite subgroup of GL(V) leaves invariant a non-degenerate Hermitean form. Thus, there is no loss in assuming that a reflection group W leaves such a form invariant.

1.2. Invariants. Let V be a finite-dimensional vector space over K ⩽ ℂ. Let K[V] denote the algebra of symmetric functions on V, i.e., the symmetric algebra S(V^{∗}) of the dual space V^{∗} of V. So K[V] is a commutative algebra over K with a grading K[V] = ⊕_{d≥0}K[V]^{d}, where, for any d ≥ 0, K[V]^{d} denotes the dth symmetric power of V^{∗}. If W ⩽ GL(V) then W acts naturally on K[V], respecting the grading. Now reflection groups are characterised by the structure of their invariant ring K[V]^{W}:

1.3. Theorem (Shephard–Todd [166], Chevalley [46]). Let V be a finite-dimen-sional vector space over a field of characteristic 0 and W ⩽ GL(V) a finite group. Then the following are equivalent:

(i) the ring of invariants K[V]^{W} is a polynomial ring, (ii) W is generated by reflections.

The implication from (i) to (ii) is an easy consequence of Molien’s formula

P (K[V]^{W} , x) = \frac{1}{|W |} ∑_{g∈W} \frac{1}{det_{V} (1 − gx)}

for the Hilbert series P (K[V]^{W} , x) of the ring of invariants K[V]^{W}, see Shephard– Todd [166, p.289]. It follows from Auslander’s purity of the branch locus that this implication remains true in arbitrary characteristic (see Benson [7, Th. 7.2.1], for example). The other direction was proved by Shephard and Todd as an application of their classification of complex reflection groups (see Section (1.13)). Chevalley gave a general proof avoiding the classification which uses the combinatorics of differential operators.

Let W be an n-dimensional reflection group. By Theorem 1.3 the ring of invariants is generated by n algebraically independent polynomials (so-called basic invariants), which may be taken to be homogeneous. Although these polynomials are not uniquely determined in general, their degrees d_{1} ⩽ … ⩽ d_{n} are. They are called the degrees of W. Then |W | = d_{1} ⋯ d_{n}, and the Molien formula shows that N(W) := ∑_{i=1}^{n} m_{i} is the number of reflections in W, where m_{i} := d_{i} − 1 are the exponents of W.

The quotient K[V]_{W} of K[V] by the ideal generated by the invariants of strictly positive degree is called the coinvariant algebra of (V, W). This is again a naturally graded W -module, whose structure is described by:

1.4. Theorem (Chevalley [46]). Let V be a finite-dimensional vector space over a field K of characteristic 0 and W ⩽ GL(V) a reflection group. Then K[V]_{W} carries a graded version of the regular representation of W. The grading is such that

∑_{i≥0} dim K[V]_{W}^{i} x^{i} = ∏_{i=1}^{n} \frac{x^{d_{i}} − 1}{x − 1},

where K[V]_{W}^{i} denotes the homogeneous component of degree i.

(See also Bourbaki [25, V.5.2, Th. 2].) The polynomial

P_{W} := ∑_{i≥0} dim K[V]_{W}^{i} x^{i}

is called the Poincaré-polynomial of W. 1.5. Parabolic subgroups. Let W be a reflection group on V. The parabolic subgroups of W are by definition the pointwise stabilisers

W_{V′} := {g ∈ W | g · v = v for all v ∈ V′}.

of subspaces V′ ⩽ V. The following result is of big importance in the theory of reflection groups:

1.6. Theorem (Steinberg [174]). Let W ⩽ GL(V) be a complex reflection group. For any subspace V′ ⩽ V the parabolic subgroup W_{V′} is generated by the reflections it contains, that is, by the reflections whose reflecting hyperplane contains V′. In particular, parabolic subgroups are themselves reflection groups.

For the proof, Steinberg characterises reflection groups via eigenfunctions of differential operators with constant coefficients that are invariant under finite linear groups. For a generalisation to positive characteristic see Theorem 5.4. 1.7. Exponents, coexponents and fake degrees. Let W be a complex reflection group. For w ∈ W define k(w) := dim V^{⟨w⟩}, the dimension of the fixed space of w on V. Solomon [168] proved the following remarkable formula for the generating function of k

∑_{w∈W} x^{k(w)} = ∏_{i=1}^{n}(x + m_{i}),

by showing that the algebra of W -invariant differential forms with polynomial coefficients is an exterior algebra of rank n over the algebra K[V]^{W}, generated by the differentials of a set of basic invariants (see also Flatto [79], Benson [7, Th. 7.3.1]). The formula was first observed by Shephard–Todd [166, 5.3] using their classification of irreducible complex reflection groups. Dually, Orlik–Solomon [153] showed

∑_{w∈W} det_{V} (w) x^{k(w)} = ∏_{i=1}^{n}(x − m_{i}^{∗})

for some non-negative integers m_{1}^{∗} ⩽ … ⩽ m_{n}^{∗}, the coexponents of W (see Lehrer– Michel [129] for a generalization, and Kusuoka [123], Orlik–Solomon [154, 155] for versions over finite fields).

Let χ be an irreducible character of W. The fake degree of χ is the polynomial

R_{χ} := ∑_{d⩾0}⟨K[V]_{W}^{d} , χ^{∗}⟩_{W} x^{d} = \frac{1}{|W |} ∑_{w∈W} \frac{χ(w)}{det_{V} (xw − 1)^{∗}} ∏_{i=1}^{n}(x^{d_{i}} − 1) ∈ ℤ[x],

that is, the graded multiplicity of χ^{∗} in the W -module K[V]_{W}. Thus, in particular, R_{χ} specialises to the degree χ(1) at x = 1. The exponents (e_{i}(χ) | 1 ⩽ i ⩽ χ(1)) of an irreducible character χ of W are defined by the formula R_{χ} = ∑_{i=1}^{χ(1)} x^{e_{i}(χ)}. The exponents m_{i} of W are now just the exponents of the contragradient of the reflection representation ρ^{∗} := tr_{V^{∗}}, that is, R_{ρ^{∗}} = ∑_{i=1}^{n} x^{m_{i}}. Dually, the coexponents are the exponents of ρ. In particular, for real reflection groups exponents and coexponents coincide. In general N^{∗}(W) := ∑_{i=1}^{n} m_{i}^{∗} equals the number of reflecting hyperplanes of W. The d_{i}^{∗} := m_{i}^{∗} − 1 are sometimes called the codegrees of W.

If W is a Weyl group (see Section 2.10), the fake degrees constitute a first approximation to the degrees of principal series unipotent characters of finite groups of Lie type with Weyl group W. See also Section 4.8 for further properties.

1.8. Reflection data. In the general theory of finite groups of Lie type (where an algebraic group comes with an action of a Frobenius map) as well as in the study of Levi subgroups it is natural to consider reflection groups together with an automorphism ϕ normalising the reflection representation; see the survey article Broué–Malle [36]. This leads to the following abstract definition.

A pair (V, W ϕ) is called a reflection datum if V is a vector space over a subfield K ⊆ ℂ and W ϕ is a coset in GL(V) of a reflection group W ⊆ GL(V), where ϕ ∈ GL(V) normalises W.

A sub-reflection datum of a reflection datum (V, W ϕ) is a reflection datum of the form (V′, W′(wϕ)|_{V′}), where V′ is a subspace of V, W′ is a reflection subgroup of N_{W} (V′)|_{V′} stabilising V′ (hence, a reflection subgroup of N_{W} (V′)/W_{V′}), and wϕ is an element of W ϕ stabilising V′ and normalising W′. A Levi sub-reflection datum of (V, W ϕ) is a sub-reflection datum of the form (V, W_{V′}(wϕ)) for some subspace V′ ⩽ V (note that, by Theorem 1.6, W_{V′} is indeed a reflection subgroup of W). A torus of 𝔾 is a sub-reflection datum with trivial reflection group.

Let 𝔾 = (V, W ϕ) be a reflection datum. Then ϕ acts naturally on the symmetric algebra K[V]. It is possible to choose basic invariants f_{1}, …, f_{n} ∈ K[V]^{W}, such that f_{i}^{ϕ} = ϵ_{i}f_{i} for roots of unity ϵ_{1}, …, ϵ_{n}. The multiset {(d_{i}, ϵ_{i})} of generalised degrees of 𝔾 then only depends on W and ϕ (see for example Springer [169, Lemma 6.1]). The polynomial order of the reflection datum 𝔾 = (V, W ϕ) is by definition the polynomial

|𝔾| := \frac{ϵ_{𝔾}x^{N(W)}}{\frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)^{∗}}} = x^{N(W)} ∏_{i=1}^{n}(x^{d_{i}} − ϵ_{i}),

where ϵ_{𝔾} := (−1)^{n}ϵ_{1} ⋯ ϵ_{n}. Let Φ(x) be a cyclotomic polynomial over K. A torus 𝕋 = (V′, (wϕ)|_{V′}) of 𝔾 is called a Φ-torus if the polynomial order of 𝕋 is a power of Φ.

Reflection data can be thought of as the skeletons of finite reductive groups.

1.9. Regular elements. In this section we present results which show that certain subgroups respectively subquotients of reflection groups are again reflection groups. Let (V, wϕ) be a reflection datum over K = ℂ. For wϕ ∈ W ϕ and a root of unity ζ ∈ ℂ^{×} write

V (wϕ, ζ) := {v ∈ V | wϕ · v = ζv}

for the ζ-eigenspace of wϕ. Note that (V (wϕ, ζ), wϕ) is an (x − ζ)-torus of (V, wϕ) in the sense defined above. These (x − ζ)-tori for fixed ζ satisfy a kind of Sylow theory. Let f_{1}, ⋯, f_{n} be a set of basic invariants for W and H_{i} the surface defined by f_{i} = 0. Springer [169] proves that

⋃_{wϕ∈W ϕ} V (wϕ, ζ) = ⋂_{i:ϵ_{i}ζ^{d_{i}} =1} H_{i} ,

the irreducible components of this algebraic set are just the maximal V (wϕ, ζ), W acts transitively on these components, and their common dimension is just the number a(d, ϕ) of indices i such that ϵ_{i}ζ^{d_{i}} = 1, where d denotes the order of ζ. (Note that a(d, ϕ) only depends on d, not on ζ itself.) From this he obtains:

1.10. Theorem (Springer [169, Th. 3.4 and 6.2]). Let (V, W ϕ) be a reflection datum over ℂ, ζ a primitive dth root of unity. Then:

(i) max{dim V (wϕ, ζ) | w ∈ W } = a(d, ϕ).

(ii) For any w ∈ W there exists a w′ ∈ W such that V (wϕ, ζ) ⊆ V (w′ϕ, ζ) and V (w′ϕ, ζ) has maximal dimension.

(iii) If dim V (wϕ, ζ) = dim V (w′ϕ, ζ) = a(d, ϕ) then there exists a u ∈ W with u · V (wϕ, ζ) = V (w′ϕ, ζ).

This can be rephrased as follows: Let K be a subfield of ℂ, Φ a cyclotomic polynomial over K. A torus 𝕋 of 𝔾 is called a Φ-Sylow torus, if its order equals the full Φ-part of the order of 𝔾. Then Φ-tori of 𝔾 satisfy the three statements of Sylow’s theorem. (For an analogue of the statement on the number of Sylow subgroups see Broué–Malle–Michel [37, Th. 5.1(4)]).

This can in turn be used to deduce a Sylow theory for tori in finite groups of Lie type (see Broué–Malle [34]).

A vector v ∈ V is called regular (for W) if it is not contained in any reflecting hyperplane, i.e. (by Theorem 1.6), if its stabiliser W_{v} is trivial. Let ζ ∈ ℂ be a root of unity. An element wϕ ∈ W ϕ is ζ-regular if V (wϕ, ζ) contains a regular vector. By definition, if wϕ is regular for some root of unity, then so is any power of wϕ. If ϕ = 1, the Theorem 1.6 of Steinberg implies that the orders of w and ζ coincide. An integer d is a regular number for W if it is the order of a regular element of W.

1.11. Theorem (Springer [169, Th. 6.4 and Prop. 4.5]). Let wϕ ∈ W ϕ be ζ-regular of order d. Then:

(i) dim V (wϕ, ζ) = a(d, ϕ).

(ii) The centraliser of wϕ in W is isomorphic to a reflection group in V (wϕ, ζ)

whose degrees are the d_{i} with ϵ_{i}ζ^{d_{i}} = 1. (iii) The elements of W ϕ with property (i) form a single conjugacy class under

W. (iv) Let ϕ = 1 and let χ be an irreducible character of W. Then the eigenvalues

of w in a representation with character χ are (ζ^{e_{i}(χ)} | 1 ⩽ i ⩽ χ(1)).

In particular, it follows from (iv) that the eigenvalues of a ζ-regular element w on V are (ζ^{m_{i}^{∗}} | 1 ⩽ i ⩽ n).

Interestingly enough, the normaliser modulo centraliser of arbitrary Sylow tori of reflection data are naturally reflection groups, as the following generalisation of the previous result shows:

1.12. Theorem (Lehrer–Springer [130, 131]). Let w ∈ W and Ṽ := V (wϕ, ζ) be such that (Ṽ , wϕ) is a Φ-Sylow torus. Let N := {w′ ∈ W | w′ · Ṽ = Ṽ } be the normaliser, C := {w′ ∈ W | w′ · v = v for all v ∈ Ṽ } the centraliser of Ṽ.

(i) Then N/C acts as a reflection group on Ṽ, with reflecting hyperplanes the intersections with Ṽ of those of W.

(ii) A set of basic invariants of N/C is given by the restrictions to Ṽ of those

f_{i} with ϵ_{i}ζ^{d_{i}} = 1. (iii) If W is irreducible on V, then so is N/C on Ṽ.

In the case of regular elements, the second assertion of (i) goes back to Lehrer [128, 5.8], Denef–Loeser [64]; see also Broué–Michel [39, Prop. 3.2]. 1.13. The Shephard–Todd classification. Let V be a finite-dimensional complex vector space and W ⩽ GL(V) a reflection group. Since W is finite, the representation on V is completely reducible, and W is the direct product of irreducible reflection subgroups. Thus, in order to determine all reflection groups over ℂ, it is sufficient to classify the irreducible ones. This was achieved by Shephard and Todd [166].

To describe this classification, first recall that a subgroup W ⩽ GL(V) is called imprimitive if there exists a direct sum decomposition V = V_{1} ⊕…⊕V_{k} with k > 1 stabilised by W (that is, W permutes the summands). The bulk of irreducible complex reflection groups consists of imprimitive ones. For any d, e, n ⩾ 1 let G(de, e, n) denote the group of monomial n × n-matrices (that is, matrices with precisely one non-zero entry in each row and column) with non-zero entries in the set of deth roots of unity, such that the product over these entries is a dth root of unity.

Explicit generators may be chosen as follows: G(d, 1, n) is generated on ℂ^{n} with standard Hermitean form) by the reflection t_{1} of order d with root the first standard basis vector b_{1} and by the permutation matrices t_{2}, …, t_{n} for the transpositions (1, 2), (2, 3), …, (n−1, n). For d > 1 this is an irreducible reflection group, isomorphic to the wreath product C_{d}≀𝔖_{n} of the cyclic group of order d with the symmetric group 𝔖_{n}, where the base group is generated by the reflections of order d with roots the standard basis vectors, and a complement consists of all permutation matrices.

Let γ_{d} : G(d, 1, n) ⟶ ℂ^{×} be the linear character of G(d, 1, n) obtained by tensoring the determinant on V with the sign character on the quotient 𝔖_{n}. Then for any e > 1 we have

G(de, e, n) := ker(γ_{de}^{d}) ⩽ G(de, 1, n).

This is an irreducible reflection subgroup of G(de, 1, n) for all n ⩾ 2, d ⩾ 1, e ⩾ 2, except for (d, e, n) = (2, 2, 2). It is generated by the reflections

t_{1}^{e}, t_{1}^{−1}t_{2}t_{1}, t_{2}, t_{3}, …, t_{n},

where the first generator is redundant if d = 1. Clearly, G(de, e, n) stabilises the decomposition V = ℂb_{1} ⊕ … ⊕ ℂb_{n} of V, so it is imprimitive for n > 1. The order of G(de, e, n) is given by d^{n}e^{n−1}n!. Using the wreath product structure it is easy to show that the only isomorphisms among groups in this series are G(2, 1, 2) ≅ G(4, 4, 2), and G(de, e, 1) ≅ G(d, 1, 1) for all d, e, while G(2, 2, 2) is reducible. All these are isomorphisms of reflection groups.

In its natural action on ℚ^{n} the symmetric group 𝔖_{n} stabilises the 1-dimensional subspace consisting of vectors with all coordinates equal and the (n−1)-dimensional subspace consisting of those vectors whose coordinates add up to 0. In its action on the latter, 𝔖_{n} is an irreducible and primitive reflection group. The classification result may now be stated as follows (see also Cohen [50]):

1.14. Theorem (Shephard–Todd [166]). The irreducible complex reflection groups are the groups G(de, e, n), for de ⩾ 2, n ⩾ 1, (de, e, n) ≠ (2, 2, 2), the groups 𝔖_{n} (n ⩾ 2) in their (n − 1)-dimensional natural representation, and 34 further primitive groups.

Moreover, any irreducible n-dimensional complex reflection group has a generating set of at most n + 1 reflections.

The primitive groups are usually denoted by G_{4}, …, G_{37}, as in the original article [166] (where the first three indices were reserved for the families of imprimitive groups G(de, e, n) (de, n ⩾ 2), cyclic groups G(d, 1, 1) and symmetric groups 𝔖_{n+1}). An n-dimensional irreducible reflection groups generated by n of its reflections is called well-generated. The groups for which this fails are the imprimitive groups G(de, e, n), d, e, n > 1, and the primitive groups

G_{i} with i ∈ {7, 11, 12, 13, 15, 19, 22, 31}.

By construction G(de, e, n) contains the well-generated group G(de, de, n). More generally, the classification implies the following, for which no a priori proof is known:

1.15. Corollary. Any irreducible complex reflection group W ⩽ GL(V) contains a well-generated reflection subgroup W′ ⩽ W which is still irreducible on V.

The primitive groups G_{4}, …, G_{37} occur in dimensions 2 up to 8. In Table 1 we collect some data on the irreducible complex reflection groups. (These and many more data for complex reflection groups have been implemented by Jean Michel into the CHEVIE-system [89].) In the first part, the dimension is always equal to n, in the second it can be read of from the number of degrees. We give the degrees, the codegrees in case they are not described by the following Theorem 1.16 (that is, if W is not well-generated), and the character field K_{W} of the reflection representation. For the exceptional groups we also give the structure of W/Z(W) and we indicate the regular degrees by boldface (that is, those degrees which are regular numbers for W, see Cohen [50, p. 395 and p. 412] and Springer [169, Tables 1–6]). The regular degrees for the infinite series are: n, n + 1 for 𝔖_{n+1}, dn for G(de, e, n) with d > 1, (n−1)e for G(e, e, n) with n|e, and (n−1)e, n for G(e, e, n) with n |̸e. Lehrer and Michel [129, Th. 3.1] have shown that an integer is a regular number if and only if it divides as many degrees as codegrees.

Fundamental invariants for most types are given in Shephard–Todd [166] as well as defining relations and further information on parabolic subgroups (see also Coxeter [56], Shephard [165] and Broué–Malle–Rouquier [38] for presentations, and the tables in Cohen [50] and Broué–Malle–Rouquier [38, Appendix 2]).

If the irreducible reflection group W has an invariant of degree 2, then it leaves invariant a non-degenerate quadratic form, so the representation may be realised over the reals. Conversely, if W is a real reflection group, then it leaves invariant a quadratic form. Thus the real irreducible reflection groups are precisely those with d_{1} = 2, that is, the infinite series G(2, 1, n), G(2, 2, n), G(e, e, 2) and 𝔖_{n+1}, and the six exceptional groups G_{23}, G_{28}, G_{30}, G_{35}, G_{36}, G_{37} (see Section 2.5).

The degrees and codegrees of a finite complex reflection group satisfy some remarkable identities. As an example, let us quote the following result, for which at present only a case-by-case proof is known:

1.16. Theorem (Orlik–Solomon [153]). Let W be an irreducible complex reflection group in dimension n. Then the following are equivalent:

(i) d_{i} + d_{n−i+1}^{∗} = d_{n} for i = 1, …, n, (ii) N + N^{∗} = nd_{n}, (iii) d_{i}^{∗} < d_{n} for i = 1, …, n, (iv) W is well-generated.

See also Terao–Yano [179] for a partial explanation.

From the Shephard–Todd classification, it is straightforward to obtain a classification of reflection data. An easy argument allows to reduce to the case where W acts irreducibly on V. Then either up to scalars ϕ can be chosen to be a reflection, or W = G_{28} is the real reflection group of type F_{4}, and ϕ induces the graph automorphism on the F_{4}-diagram (see Broué–Malle–Michel [37, Prop. 3.13]). An infinite series of examples is obtained from the embedding of G(de, e, n) into G(de, 1, n) (which is the full projective normaliser in all but finitely many cases). Apart from this, there are only six further cases, which we list in Table 2.

CONTRIBUTION TO THE HANDBOOK OF ALGEBRA 9

Table 1. Irreducible complex reflection groups Infinite series

W  degrees  codegrees  K_{W}
G(d, 1, n) (d⩾2, n⩾1) d, 2d, …, nd  ∗  ℚ(ζ_{d})
G(de, e, n) (d, e, n⩾2) ed, 2ed, …, (n−1)ed, nd 0, ed, …, (n−1)ed ℚ(ζ_{de})
G(e, e, n) (e⩾2, n⩾3) e, 2e, …, (n−1)e, n  ∗  ℚ(ζ_{e})
G(e, e, 2) (e⩾3)  2, e  ∗  ℚ(ζ_{e}+ζ_{e}^{−1})
𝔖_{n+1} (n⩾1)  2, 3, …, n+1  ∗  ℚ

Exceptional groups

W degrees  codegrees K_{W}  W/Z(W)
G_{4} 4, 6  ∗  ℚ(ζ_{3})  𝔄_{4}
G_{5} 6, 12  ∗  ℚ(ζ_{3})  𝔄_{4}
G_{6} 4, 12  ∗  ℚ(ζ_{12})  𝔄_{4}
G_{7} 12, 12  0, 12  ℚ(ζ_{12})  𝔄_{4}
G_{8} 8, 12  ∗  ℚ(i)  𝔖_{4}
G_{9} 8, 24  ∗  ℚ(ζ_{8})  𝔖_{4}
G_{10} 12, 24  ∗  ℚ(ζ_{12})  𝔖_{4}
G_{11} 24, 24  0, 24  ℚ(ζ_{24})  𝔖_{4}
G_{12} 6, 8  0, 10  ℚ(\sqrt{−2})  𝔖_{4}
G_{13} 8, 12  0, 16  ℚ(ζ_{8})  𝔖_{4}
G_{14} 6, 24  ∗  ℚ(ζ_{3}, \sqrt{−2})  𝔖_{4}
G_{15} 12, 24  0, 24  ℚ(ζ_{24})  𝔖_{4}
G_{16} 20, 30  ∗  ℚ(ζ_{5})  𝔄_{5}
G_{17} 20, 60  ∗  ℚ(ζ_{20})  𝔄_{5}
G_{18} 30, 60  ∗  ℚ(ζ_{15})  𝔄_{5}
G_{19} 60, 60  0, 60  ℚ(ζ_{60})  𝔄_{5}
G_{20} 12, 30  ∗  ℚ(ζ_{3}, \sqrt{5})  𝔄_{5}
G_{21} 12, 60  ∗  ℚ(ζ_{12}, \sqrt{5})  𝔄_{5}
G_{22} 12, 20  0, 28  ℚ(i, \sqrt{5})  𝔄_{5}
G_{23} 2, 6, 10  ∗  ℚ(\sqrt{5})  𝔄_{5}
G_{24} 4, 6, 14  ∗  ℚ(\sqrt{−7})  GL_{3}(2)
G_{25} 6, 9, 12  ∗  ℚ(ζ_{3})  3^{2} : SL_{2}(3)
G_{26} 6, 12, 18  ∗  ℚ(ζ_{3})  3^{2} : SL_{2}(3)
G_{27} 6, 12, 30  ∗  ℚ(ζ_{3}, \sqrt{5})  𝔄_{6}
G_{28} 2, 6, 8, 12  ∗  ℚ  2^{4} : (𝔖_{3} × 𝔖_{3})
G_{29} 4, 8, 12, 20  ∗  ℚ(i)  2^{4} : 𝔖_{5}
G_{30} 2, 12, 20, 30  ∗  ℚ(\sqrt{5})  𝔄_{5} ≀ 2
G_{31} 8, 12, 20, 24  0, 12, 16, 28 ℚ(i)  2^{4} : 𝔖_{6}
G_{32} 12, 18, 24, 30  ∗  ℚ(ζ_{3})  U_{4}(2)
G_{33} 4, 6, 10, 12, 18  ∗  ℚ(ζ_{3})  O_{5}(3)
G_{34} 6, 12, 18, 24, 30, 42  ∗  ℚ(ζ_{3})  O_{6}^{−}(3).2
G_{35} 2, 5, 6, 8, 9, 12  ∗  ℚ  O_{6}^{−}(2)
G_{36} 2, 6, 8, 10, 12, 14, 18  ∗  ℚ  O_{7}(2)
G_{37} 2, 8, 12, 14, 18, 20, 24, 30 ∗  ℚ  O_{8}^{+}(2).2

Table 2. Exceptional twisted reflection data

W  d_{i}  ϵ_{i}  field  origin of ϕ
G(4, 2, 2) 4, 4  1, ζ_{3}  ℚ(i)  < G_{6}
G(3, 3, 3) 3, 6, 3  1, 1, −1  ℚ(ζ_{3})  < G_{26}
G(2, 2, 4) 2, 4, 4, 6 1, ζ_{3}, ζ_{3}^{2}, 1 ℚ  < G_{28}
G_{5}  6, 12  1, −1  ℚ(ζ_{3}, \sqrt{−2}) < G_{14}
G_{7}  12, 12  1, −1  ℚ(ζ_{12})  < G_{10}
G_{28}  2, 6, 8, 12 1, −1, 1, −1 ℚ(\sqrt{2})  graph aut.

2. Real reflection groups

In this section we discuss in more detail the special case where W is a real reflection group. This is a well-developped theory, and there are several good places to learn about real reflection groups: the classical Bourbaki volume [25], the very elementary text by Benson–Grove [8], the relevant chapters in Curtis–Reiner [58], Hiller [99], and Humphreys [105]. Various pieces of the theory have also been recollected in a concise way in articles by Steinberg [177]. The exposition here partly follows Geck–Pfeiffer [95, Chap. 1]. We shall only present the most basic results and refer to the above textbooks and our bibliography for further reading. 2.1. Coxeter groups. Let S be a finite non-empty index set and M = (m_{st})_{s,t∈S} be a symmetric matrix such that m_{ss} = 1 for all s ∈ S and m_{st} ∈ {2, 3, 4, …}∪{∞} for all s ≠ t in S. Such a matrix is called a Coxeter matrix. Now let W be a group containing S as a subset. (W may be finite or infinite.) Then the pair (W, S) is called a Coxeter system, and W is called a Coxeter group, if W has a presentation with generators S and defining relations of the form

(st)^{m_{st}} = 1 for all s, t ∈ S with m_{st} < ∞;

in particular, this means that s^{2} = 1 for all s ∈ S. Therefore, the above relations (for s ≠ t) can also be expressed in the form

\underbrace{sts ⋯}_{m_{st} times} = \underbrace{tst ⋯}_{m_{st} times} for all s, t ∈ S with 2 ⩽ m_{st} < ∞.

We say that C is of finite type and that (W, S) is a finite Coxeter system if W is a finite group. The information contained in M can be visualised by a corresponding Coxeter graph, which is defined as follows. It has vertices labelled by the elements of S, and two vertices labelled by s ≠ t are joined by an edge if m_{st} ⩾ 3. Moreover, if m_{st} ⩾ 4, we label the edge by m_{st}. The standard example of a finite Coxeter system is the pair (𝔖_{n}, {s_{1}, …, s_{n−1}}) where s_{i} = (i, i + 1) for 1 ⩽ i ⩽ n − 1. The corresponding graph is

[figure: A_{n−1}; s_{1}; s_{2}; s_{3}; s_{n−1}]

Coxeter groups have a rich combinatorial structure. A basic tool is the length function l: W → ℕ_{0}, which is defined as follows. Let w ∈ W. Then l(w) is the length of a shortest possible expression w = s_{1} ⋯ s_{k} where s_{i} ∈ S. An expression of w of length l(w) is called a reduced expression for w. We have l(1) = 0 and l(s) = 1 for s ∈ S. Here is a key result about Coxeter groups.

2.2. Theorem (Matsumoto [140]; see also Bourbaki [25]). Let (W, S) be a Coxeter system and M be a monoid, with multiplication ⋆: M × M → M. Let f: S → M be a map such that

\underbrace{f(s) ⋆ f(t) ⋆ f(s) ⋆ ⋯}_{m_{st} times} = \underbrace{f(t) ⋆ f(s) ⋆ f(t) ⋆ ⋯}_{m_{st} times}

for all s ≠ t in S such that m_{st} < ∞. Then there exists a unique map F: W → M such that F (w) = f(s_{1}) ⋆ ⋯ ⋆ f(s_{k}) whenever w = s_{1} ⋯ s_{k} (s_{i} ∈ S) is reduced.

Typically, Matsumoto’s theorem can be used to show that certain constructions with reduced expressions of elements of W actually do not depend on the choice of the reduced expressions. We give two examples.

(1) Let w ∈ W and take a reduced expression w = s_{1} ⋯ s_{k} with s_{i} ∈ S. Then the set {s_{1}, …, s_{k}} does not depend on the choice of the reduced expression.

(Indeed, let M be the monoid whose elements are the subsets of S and product given by A ⋆ B := A ∪ B. Then the assumptions of Matsumoto’s theorem are satisfied for the map f: S → M, s ↦ {s}, and this yields the required assertion.)

(2) Let w ∈ W and fix a reduced expression w = s_{1} ⋯ s_{k} (s_{i} ∈ S). Consider the set of all subexpressions:

S(w) := {y ∈ W | y = s_{i_{1}} ⋯ s_{i_{l}}where l ⩾ 0 and 1 ⩽ i_{1} < ⋯ < i_{l} ⩽ k}.

Then S(w) does not depend on the choice of the reduced expression for w.

(Indeed, let M be the monoid whose elements are the subsets of W and product given by A ⋆ B := {ab | a ∈ A, b ∈ B} (for A, B ⊆ W). Then the assumptions of Matsumoto’s theorem are satisfied for the map f: S → M, s ↦ {1, s}, and this yields the required assertion.)

We also note that the so-called Exchange Condition and the Cancellation Law are further consequences of the above results. The “Cancellation Law” states that, given w ∈ W and an expression w = s_{1} ⋯ s_{k} (s_{i} ∈ S) which is not reduced, one can obtain a reduced expression of w by simply cancelling some of the factors in the given expression. This law together with (2) yields that the relation

y ⩽ w ⟺^{def} y ∈ S(w)

is a partial order on W, called the Bruhat–Chevalley order. This ordering has been extensively studied; see, for example, Verma [183], Deodhar [65], Björner [19], Lascoux–Schützenberger [126] and Geck–Kim [90]. By Chevalley [48], it is related to the Bruhat decomposition in algebraic groups; we will explain this result in (2.16) below. 2.3. Cartan matrices. Let M = (m_{st})_{s,t∈S} be a Coxeter matrix as above. We can also associate with M a group generated by reflections. This is done as follows. Choose a matrix C = (c_{st})_{s,t∈S} with entries in ℝ such that the following conditions are satisfied:

(C1) For s ≠ t we have c_{st} ⩽ 0; furthermore, c_{st} ≠ 0 if and only if c_{ts} ≠ 0.

(C2) We have c_{ss} = 2 and, for s ≠ t, we have c_{st}c_{ts} = 4 cos^{2}(π/m_{st}). Such a matrix C will be called a Cartan matrix associated with M. For example, we could simply take c_{st} := −2 cos(π/m_{st}) for all s, t ∈ S; this may be called the standard Cartan matrix associated with M. We always have 0 ⩽ c_{st}c_{ts} ⩽ 4. Here are some values for the product c_{st}c_{ts}:

\begin{matrix}m_{st} & 2 & 3 & 4 & 5 & 6 & 8 & ∞ \\ c_{st}c_{ts} & 0 & 1 & 2 & (3 + \sqrt{5})/2 & 3 & 2 + \sqrt{2} & 4 \end{matrix}

Now let V be an ℝ-vector space of dimension |S|, with a fixed basis {α_{s} | s ∈ S}. We define a linear action of the elements in S on V by the rule:

s: V → V, α_{t} ↦ α_{t} − c_{st}α_{s} (t ∈ S).

It is easily checked that s ∈ GL(V) has order 2 and precisely one eigenvalue −1 (with eigenvector α_{s}). Thus, s is a reflection with root α_{s}. We then define

W = W (C) := ⟨S⟩ ⊂ GL(V);

thus, if |W | < ∞ is finite, then W will be a real reflection group. Now we can state the following basic result.

2.4. Theorem (Coxeter [54, 55]). Let M = (m_{st})_{s,t∈S} be a Coxeter matrix and C be a Cartan matrix associated with M. Let W (C) = ⟨S⟩ ⊆ GL(V) be the group constructed as in (2.3). Then the pair (W (C), S) is a Coxeter system. Furthermore, the group W (C) is finite if and only if

(∗)  the matrix (− cos(π/m_{st}))_{s,t∈S} is positive-definite,

i.e., we have det(− cos(π/m_{st}))_{s,t∈J} > 0 for every subset J ⊆ S. All finite real reflection groups arise in this way.

The fact that (W (C), S) is a Coxeter system is proved in [95, 1.2.7]. The finiteness condition can be found in Bourbaki [25, Chap. V, §4, no. 8]. Finally, the fact that all finite real reflection groups arise in this way is established in [25, Chap. V, §3, no. 2]. The “note historique” in [25] contains a detailed account of the history of the above result. 2.5. Classification of finite Coxeter groups. (See also Section 1.13.) Let M = (m_{st})_{s,t∈S} be a Coxeter matrix. We say that M is decomposable if there is a partition S = S_{1} ⨿ S_{2} with S_{1}, S_{2} ≠ ∅ and such that m_{st} = 2 whenever s ∈ S_{1}, t ∈ S_{2}. If C = (c_{st}) is any Cartan matrix associated with M, then this condition translates to: c_{st} = c_{ts} = 0 whenever s ∈ S_{1}, t ∈ S_{2}. Correspondingly, we also have a direct sum decomposition V = V_{1} ⊕ V_{2} where V_{1} has basis {α_{s} | s ∈ S_{1}} and V_{2} has basis {α_{s} | s ∈ S_{2}}. Then it easily follows that we have an isomorphism

W (C) ⟶^{∼} W (C_{1}) × W (C_{2}), w ↦ (w|_{V_{1}}, w|_{V_{2}}).

In this way, the study of the groups W (C) is reduced to the case where C is indecomposable (i.e., there is no partition S = S_{1} ⨿ S_{1} as above). If this holds, we call the corresponding Coxeter system (W, S) an irreducible Coxeter system.

2.6. Theorem. The Coxeter graphs of the indecomposable Coxeter matrices M such that condition (∗) in Theorem 2.4 holds are precisely the graphs in Table 3.

For the proof of this classification, see [25, Chap. VI, no. 4.1]. The identification with the groups occurring in the Shephard–Todd classification (see Theorem 1.14)

Table 3. Coxeter graphs of irreducible finite Coxeter groups

[figure: E_{7}; 1; 3; 4; 2; 5; 6; 7; E_{8}; 1; 3; 4; 2; 5; 6; 7; 8; I_{2}(m); m⩾5; 1; m; 2; F_{4}; 1; 2; 4; 3; 4; H_{3}; 1; 5; 2; 3; H_{4}; 1; 5; 2; 3; 4; E_{6}; 1; 3; 4; 2; 5; 6; B_{n}; n⩾2; 1; 4; 2; 3; n; A_{n}; n⩾1; 1; 2; 3; n; D_{n}; n⩾4; 1; 2; 3; 4; n]

The numbers on the vertices correspond to a chosen labelling of the elements of S.

is given in the following table.

Coxeter graph Shephard–Todd
A_{n−1}  𝔖_{n}
B_{n}  G(2, 1, n)
D_{n}  G(2, 2, n)
I_{2}(m)  G(m, m, 2)
H_{3}  G_{23}
H_{4}  G_{30}
F_{4}  G_{28}
E_{6}  G_{35}
E_{7}  G_{36}
E_{8}  G_{37}

Thus, any finite irreducible real reflection group is the reflection group arising from a Cartan matrix associated with one of the graphs in Table 3.

Now, there are a number of results on finite Coxeter groups which can be formulated in general terms but whose proof requires a case-by-case verification using the above classification. We mention two such results, concerning conjugacy classes.

2.7. Theorem (Carter [41]). Let (W, S) be a finite Coxeter system. Then every element in W is conjugate to its inverse. More precisely, given w ∈ W, there exist x, y ∈ W such that w = xy and x^{2} = y^{2} = 1.

Every element x ∈ W such that x^{2} = 1 is a product of pairwise commuting reflections in W. Given w ∈ W and an expression w = xy as above, the geometry of the roots involved in the reflections determining x, y yields a diagram which can be used to label the conjugacy class of w. Complete lists of these diagrams can be found in [41]. 2.8. Conjugacy classes and the length function. Let (W, S) be a Coxeter system and C be a conjugacy class in W. We will be interested in studying how conjugation inside C relates to the length function on W. For this purpose, we introduce two relations, following Geck–Pfeiffer [94].

Given x, y ∈ W and s ∈ S, we write x ⟶^{s} y if y = sxs and l(y) ⩽ l(x). We shall write x ⟶ y if there are sequences x_{0}, x_{1}, …, x_{n} ∈ W and s_{1}, …, s_{n} ∈ S (for some n ⩾ 0) such that

x = x_{0} ⟶^{s_{1}} x_{1} ⟶^{s_{2}} x_{2} ⟶^{s_{3}} ⋯ ⟶^{s_{n}} x_{n} = y.

Thus, we have x ⟶ y if we can go from x to y by a chain of conjugation with generators in S such that, at each step, the length of the elements either remains the same or decreases.

In a slightly different direction, let us now consider two elements x, y ∈ W such that l(x) = l(y). We write x ∼^{w} y (where w ∈ W) if wx = yw and l(wx) = l(w)+l(x) or xw = wy and l(wy) = l(w) + l(y). We write x ∼ y if there are sequences x_{0}, x_{1}, …, x_{n} ∈ W and w_{1}, …, w_{n} ∈ W (for some n ⩾ 0) such that

x = x_{0} ∼^{w_{1}} x_{1} ∼^{w_{2}} x_{2} ∼^{w_{3}} ⋯ ∼^{w_{n}} x_{n} = y.

Thus, we have x ∼ y if we can go from x to y by a chain of conjugation with elements of W such that, at each step, the length of the elements remains the same and an additional length condition involving the conjugating elements is satisfied. This additional condition has the following significance. Consider a group M with multiplication ⋆ and assume that we have a map f: S → M which satisfies the requirements in Matsumoto’s theorem 2.2. Then we have a canonical extension of f to a map F: W → M such that F (ww′) = F (w) ⋆ F (w′) whenever l(ww′) = l(w) + l(w′). Hence, in this setting, we have

x ∼ y ⇒ F (x), F (y) are conjugate in M.

Thus, we can think of the relation “∼” as “universal conjugacy”.

2.9. Theorem (Geck–Pfeiffer [94], [95]). Let (W, S) be a finite Coxeter system and C be a conjugacy class in W. We set l_{min}(C) := min{l(w) | w ∈ C} and

C_{min} := {w ∈ C | l(w) = l_{min}(C)}.

Then the following hold:

(a) For every x ∈ C, there exists some y ∈ C_{min} such that x ⟶ y.

(b) For any two elements x, y ∈ C_{min}, we have x ∼ y.

Precursors of the above result for type A have been found much earlier by Starkey [172]; see also Ram [161]. The above result allows to define the character table of the Iwahori–Hecke algebra associated with (W, S). This is discussed in more detail in the chapter on Hecke algebras (??). See Richardson [164], Geck– Michel [93], Geck–Pfeiffer [95, Chap. 3], Geck–Kim–Pfeiffer [91] and Shi [167] for further results on conjugacy classes. Krammer [121] studies the conjugacy problem for arbitrary (infinite) Coxeter groups. 2.10. The crystallographic condition. Let M = (m_{st})_{s,t∈S} be a Coxeter matrix such that the connected components of the corresponding Coxeter graph occur in Table 3. We say that M satisfies the crystallographic condition if there exists a Cartan matrix C associated with M which has integral coefficients. In this case, the corresponding reflection group W = W (C) is called a Weyl group. The significance of this notion is that there exists a corresponding semisimple Lie algebra over ℂ; see (2.15).

Now assume that M is crystallographic. By condition (C2) this implies m_{st} ∈ {2, 3, 4, 6} for all s ≠ t. Conversely, if m_{st} satisfies this condition, then we have

c_{st}c_{ts} = 0  if m_{st} = 2,
c_{st}c_{ts} = 1  if m_{st} = 3,
c_{st}c_{ts} = 2  if m_{st} = 4,
c_{st}c_{ts} = 3  if m_{st} = 6.

Thus, in each of these cases, we see that there are only two choices for c_{st} and c_{ts}: we must have c_{st} = −1 or c_{ts} = −1 (and then the other value is determined). We encode this additional information in the Coxeter graph, by putting an arrow on the edge between the nodes labelled by s, t according to the following scheme:

Edge between s ≠ t Values for c_{st}, c_{ts}
m_{st} = 3  no arrow  c_{st} = c_{ts} = −1
m_{st} = 4  [figure: s; >; t]  c_{st} = −1 c_{ts} = −2
m_{st} = 6  [figure: s; >; t]  c_{st} = −1 c_{ts} = −3

The Coxeter graph of M equipped with this additional information will be called a Dynkin diagram; it uniquely determines an integral Cartan matrix C associated with M (if such a Cartan matrix exists). The complete list of connected components of Dynkin diagrams is given in Table 4. We see that all irreducible finite Coxeter groups are Weyl groups, except for those of type H_{3}, H_{4} and I_{2}(m) where m = 5 or m ⩾ 7. Note that type B_{n} is the only case where we have two different Dynkin diagrams associated with the same Coxeter graph.

Table 4. Dynkin diagrams of Cartan matrices of finite type

[figure: E_{7}; 1; 3; 4; 2; 5; 6; 7; E_{8}; 1; 3; 4; 2; 5; 6; 7; 8; G_{2} = I_{2}(6); 1; >; 2; F_{4}; 1; 2; >; 3; 4; E_{6}; 1; 3; 4; 2; 5; 6; D_{n}; n⩾4; 1; 2; 3; 4; n; C_{n}; n⩾2; 1; >; 2; 3; n; A_{n}; n⩾1; 1; 2; 3; n; B_{n}; n⩾3; 1; <; 2; 3; n]

The following discussion of root systems associated with finite reflection groups follows the appendix on finite reflection groups in Steinberg [175]. 2.11. Root systems. Let V be a finite dimensional real vector space and let ( , ) be a positive-definite scalar product on V. Given a non-zero vector α ∈ V, the corresponding reflection w_{α} ∈ GL(V) is defined by

w_{α}(v) = v − 2\frac{(v, α)}{(α, α)}α for all v ∈ V .

Note that, for any w ∈ GL(V), we have ww_{α}w^{−1} = w_{w(α)}. A finite subset ϕ ⊆ V \ {0} is called a root system if the following conditions are satisfied:

(R1) For any α ∈ Φ, we have Φ ∩ ℝα = {±α}; (R2) For every α, β ∈ Φ, we have w_{α}(β) ∈ Φ.

Let W (Φ) ⊆ GL(V) be the subgroup generated by the reflections w_{α} (α ∈ Φ). A subset Π ⊆ Φ is called a simple system if Π is linearly independent and if any root in Φ can be written as a linear combination of the elements of Π in which all non-zero coefficients are either all positive or all negative. It is known that simple systems always exist, and that any two simple systems can be transformed into each other by an element of W (Φ). Let now fix a simple system Π ⊆ Φ. Then it is also known that

W (Φ) = ⟨w_{α} | α ∈ Π⟩  and  (α, β) ⩾ 0 for all α ≠ β in Π.

For α ≠ β in Π, let m_{αβ} ⩾ 2 be the order of w_{α}w_{β} in GL(V). Then w_{α}w_{β} is a rotation in V through the angle 2π/m_{αβ} and that we have the relation

\frac{(α, β)}{(α, α)} \frac{(β, α)}{(β, β)} = cos^{2}(π/m_{αβ}) for all α ≠ β in Π.

Thus, if we set M := (m_{αβ})_{α,β∈Π} (where m_{αα} = 1 for all α ∈ Π) and define

C := (c_{αβ})_{α,β∈Π} where c_{αβ} := 2\frac{(α, β)}{(α, α)} for α, β ∈ S,

then M is a Coxeter matrix, C is an associated Cartan matrix and W (Φ) = W (C) is the Coxeter group with Coxeter matrix M; see Theorem 2.4. Thus, every root system leads to a finite Coxeter group.

Conversely, let (W, S) be a Coxeter system associated to a Coxeter matrix M = (m_{st})_{s,t∈S}. Let C be a corresponding Cartan matrix such that c_{st} = c_{ts} = −2 cos(π/m_{st}) for every s ≠ t in S such that m_{st} is odd. Then one can show that

Φ = Φ(C) := {w(α_{s}) | w ∈ W, s ∈ S} ⊆ V

is a root system with simple system {α_{s} | s ∈ S} and that W (Φ) = W; see, e.g., Geck–Pfeiffer [95, Chap. 1]. Thus, every Coxeter group leads to a root system.

Given a root system Φ as above, there is a strong link between the combinatorics of the Coxeter presentation of W = W (Φ) and the geometry of Φ. To state the following basic result, we set

Φ^{+} := {α ∈ Π | α = ∑_{β∈Π} x_{β}β where x_{β} ∈ ℝ_{⩾0} for all β ∈ Π};

the roots in Φ^{+} will be called positive roots. Similarly, Φ^{−} := −Φ^{+} will be called the set of negative roots. By the definition of a simple system, we have Φ = Φ^{+} ⨿ Φ^{−}.

2.12. Proposition. Given α ∈ Π and w ∈ W, we have

w^{−1}(α) ∈ Φ^{+} ⇔ l(w_{α}w) = l(w) + 1,
w^{−1}(α) ∈ Φ^{−} ⇔ l(w_{α}w) = l(w) − 1.

Furthermore, for any w ∈ W, we have l(w) = |{α ∈ Φ^{+} | w(α) ∈ Φ^{−}}|.

The root systems associated with finite Weyl groups are explicitly described in Bourbaki [25, p. 251–276]. For type H_{3} and H_{4}, see Humphreys [105, 2.13].

Bremke–Malle [26, 27] have studied suitable generalisations of root systems and length functions for the infinite series G(d, 1, n) and G(e, e, n), which have subsequently been extended in weaker form by Rampetas–Shoji [162] to arbitrary imprimitive reflection groups. For investigations of root systems see also Nebe [146] and Hughes–Morris [103]. But there is no general theory of root systems and length functions for complex reflection groups (yet). 2.13. Torsion primes. Assume that Φ ⊆ V is a root system as above, with a set of simple roots Π ⊆ Φ. Assume that the corresponding Cartan matrix C is indecomposable and has integral coefficients. Thus, its Dynkin diagram is one of the graphs in Table 4. Following Springer–Steinberg [171, §I.4] we shall now discuss “bad primes” and “torsion primes” with respect to Φ.

For every α ∈ Φ, the corresponding coroot is defined by α^{∗} := 2α/(α, α). Then Φ^{∗} := {α^{∗} | α ∈ Φ} also is a root system, the dual of Φ. The Dynkin diagram of Φ^{∗} is obtained from that of Φ by reversing the arrows. (For example, the dual of a root system of type B_{n} is of type C_{n}.)

Let L(Φ) denote the lattice spanned by Φ in V. A prime number p > 0 is called bad for Φ if L(Φ)/L(Φ_{1}) has p-torsion for some (integrally) closed subsystem Φ_{1} of Φ. The prime p is called a torsion prime if L(Φ^{∗})/L(Φ_{1}^{∗}) has p-torsion for some closed subsystem Φ_{1} of Φ. Note that Φ_{1}^{∗} need not be closed in Φ^{∗}, and so the torsion primes for Φ and the bad primes for Φ^{∗} need not be the same. The bad primes can be characterised as follows. Let α_{0} = ∑_{α∈Π} m_{α}α be the unique positive root of maximal height. (The height of a root is the sum of the coefficients in the expression of that root as a linear combination of simple roots.) Then we have:

p bad ⇔  \begin{matrix}p = m_{α} \\ for some α \end{matrix} ⇔ p divides m_{α} ; for some α ⇔  \begin{matrix}p ⩽ m_{α} \\ for some α \end{matrix} .

Now let α_{0}^{∗} = ∑_{α∈Π} m_{α}^{∗}α^{∗}. Then p is a torsion prime if and only if p satisfies one of the above conditions, with m_{α} replaced by m_{α}^{∗}. For the various roots systems, the bad primes and the torsion primes are given as follows.

Type  A_{n}  B_{n}  C_{n}  D_{n}  G_{2} F_{4} E_{6} E_{7} E_{8}
(n ⩾ 2) (n ⩾ 2) (n ⩾ 4)
Bad  none  2  2  2  2, 3 2, 3 2, 3 2, 3 2, 3, 5
Torsion none  2  none  2  2 2, 3 2, 3 2, 3 2, 3, 5

The bad primes and torsion primes play a role in various questions related to sub-root systems, centralizers of semisimple elements in algebraic groups, the classification of unipotent classes in simple algebraic groups and so on; see [171] and also the survey in [43, §§1.14–1.15].

2.14. Affine Weyl groups. Let Φ ⊆ V be a root system as above, with Weyl group W. Let L(Φ) := ∑_{α∈Π} ℤα ⊆ V be the lattice spanned by the roots in V. Then W leaves L(Φ) invariant and we have a natural group homomorphism W → Aut(L(Φ)). The semidirect product

W_{a}(Φ) := L(Φ) ⋊ W

is called the affine Weyl group associated with the root system W; see Bourbaki [25, Chap. VI, §2]. The group W_{a}(Φ) itself is a Coxeter group. The corresponding presentation can also be encoded in a graph, as follows. Let α_{0} be the unique positive root of maximal height in Φ. We define an extended Cartan matrix C̃ by similar rules as before:

c̃_{αβ} := 2\frac{(α, β)}{(α, α)} for α, β ∈ Π ∪ \{−α_{0}\}.

The extended Dynkin diagrams encoding these matrices for irreducible W are given in Table 5. They are obtained from the diagrams in Table 4 by adjoining an additional node (corresponding to −α_{0}) and putting edges according to the same rules as before.

Table 5. Extended Dynkin diagrams

[figure: Ẽ_{6}; 1; 3; 4; 2; 0; 5; 6; Ẽ_{8}; 1; 3; 4; 2; 5; 6; 7; 8; 0; F̃_{4}; 0; 1; 2; >; 3; 4; Ẽ_{7}; 0; 1; 3; 2; 4; 5; 6; 7; G̃_{2}; 0; 1; >; 2; B̃_{n}; n⩾3; 1; <; 2; 3; n−1; n; 0; D̃_{n}; n⩾4; 1; 2; 3; 4; n−1; n; 0; C̃_{n}; n⩾2; 1; >; 2; 3; n; <; 0; Ã_{n}; n⩾2; 1; 2; 3; n; 0; Ã_{1}; 1; ∞; 0]

In the following subsections, we describe some situations where Coxeter groups and root systems arise “in nature”. 2.15. Kac–Moody algebras. Here we briefly discuss how Coxeter groups and root systems arise in the theory of Lie algebras or, more generally, Kac–Moody algebras. We follow the exposition in Kac [113]. Let C = (c_{st})_{s,t∈S} be a Cartan matrix all of whose coefficients are integers. We also assume that C is symmetrisable, i.e., there exists a diagonal invertible matrix D and a symmetric matrix B such that C = DB. A realisation of C is a triple (𝔥, Π, Π^{∨}) where 𝔥 is a complex vector space, Π = {α_{s} | s ∈ S} ⊆ 𝔥^{∗} := Hom(𝔥, ℂ) and Π^{∨} = {α_{s}^{∨} | s ∈ S} are subsets of 𝔥^{∗} and 𝔥, respectively, such that the following conditions hold.

(a) Both sets Π and Π^{∨} are linearly independent;

(b) we have ⟨α_{s}^{∨}, α_{t}⟩ := α_{t}(α_{s}^{∨}) = c_{st} for all s, t ∈ S;

(c) |S| − rank(C) = dim 𝔥 − |S|. Let 𝔤(C) be the corresponding Kac–Moody algebra. Then 𝔤(C) is a Lie algebra which is generated by 𝔥 together with two collections of elements {e_{s} | s ∈ S} and {f_{s} | s ∈ S}, where the following relations hold:

[e_{s}, f_{t}] = δ_{st}α_{s}^{∨}  (s, t ∈ S),
[h, h′] = 0  (h, h′ ∈ 𝔥),
[h, e_{s}] = ⟨h, α_{s}, ⟩e_{s}  (s ∈ S, h ∈ 𝔥),
[h, f_{s}] = −⟨h, α_{s}⟩f_{s}  (s ∈ S, h ∈ 𝔥),
(ad e_{s})^{1−c_{st}}e_{t} = 0  (s, t ∈ S, s ≠ t),
(ad f_{s})^{1−c_{st}}f_{t} = 0  (s, t ∈ S, s ≠ t).

(By [113, 9.11], this is a set of defining relations for 𝔤(C).) We have a direct sum decomposition

𝔤(C) = 𝔥 ⊕ ⨁_{0≠α∈Q} 𝔤_{α}(C)  where  Q := ∑_{s∈S} ℤα_{s} ⊆ 𝔥^{∗}

and 𝔤_{α}(C) := {x ∈ 𝔤(C) | [h, x] = α(h)x for all h ∈ 𝔥} for all α ∈ Q; here, 𝔥 = 𝔤_{0}. The set of all 0 ≠ α ∈ Q such that 𝔤_{α}(C) ≠ {0} will be denoted by Φ and called the root system of 𝔤(C).

For each s ∈ S, we define a linear map σ_{s}: 𝔥^{∗} → 𝔥^{∗} by the formula

σ_{s}(λ) = λ − ⟨λ, α_{s}^{∨}⟩ α_{s} for λ ∈ 𝔥^{∗}.

Then it is easily checked that σ_{s} is a reflection where σ_{s}(α_{s}) = −α_{s}. We set

W = W (C) = ⟨σ_{s} | s ∈ S⟩ ⊆ GL(𝔥^{∗}).

Now we can state (see [113, 3.7, 3.11 and 3.13]):

(a) The pair (W, {σ_{s} | s ∈ S}) is a Coxeter system; the corresponding Coxeter matrix is the one associated to C.

(b) The root system Φ is invariant under the action of W and we have l(σ_{s}w) = l(w) + 1 if and only if w^{−1}(α_{s}) ∈ Φ^{+}, where Φ^{+} is defined as in (2.11).

Thus, W and Φ have similar properties as before. Note, however, that here we did not make any assumption on C (except that it is symmetrisable with integer entries) and so W and Φ may be infinite. The finite case is characterised as follows:

|W | < ∞ ⇔ |Φ| < ∞ ⇔ dim 𝔤(C) < ∞

⇔ all connected components of C occur in Table 4.

(This follows from [113, 3.12] and the characterisation of finite Coxeter groups in Theorem 2.4.) In fact, the finite dimensional Kac–Moody algebras are precisely the “classical” semisimple complex Lie algebras (see, for example, Humphreys [104]). The Kac–Moody algebras and the root systems associated to so-called Cartan matrices of affine type have an extremely rich structure and many applications in other branches of mathematics and mathematical physics; see Kac [113].

2.16. Groups with a BN-pair. Let G be an abstract group. We say that G is a group with a BN-pair or that G admits a Tits system if there are subgroups B, N ⊆ G such that the following conditions are satisfied. (BN1) G is generated by B and N. (BN2) T := B ∩ N is normal in N and the quotient W := N/T is a finite group

generated by a set S of elements of order 2. (BN3) n_{s}Bn_{s} ≠ B if s ∈ S and n_{s} is a representative of s in N. (BN4) n_{s}Bn ⊆ Bn_{s}nB ∪ BnB for any s ∈ S and n ∈ N.

The group W is called the Weyl group of G. In fact, it is a consequence of the above axioms that the pair (W, S) is a Coxeter system; see [25, Chap. IV, §2, Théorème 2]. The notion of groups with a BN-pair has been invented by Tits; see [180]. The standard example of a group with a BN-pair is the general linear group G = GL_{n}(K), where K is any field and

B := subgroup of all upper triangular matrices in G,
N := subgroup of all monomial matrices in G.

(A matrix is called monomial if it has exactly one non-zero entry in each row and and each column.) We have

T := B ∩ N = subgroup of all diagonal matrices in G

and W = N/T ≅ 𝔖_{n}, Thus, 𝔖_{n} is the Weyl group of G. More generally, the Chevalley groups (and their twisted analogues) associated with the semisimple complex Lie algebras all have BN-pairs; see Chevalley [45], Carter [42] and Steinberg [175].

The above set of axioms imposes very strong conditions on the structure of a group G with a BN-pair. For example, we have the following Bruhat decomposition, which gives the decomposition of G into double cosets with respect to B:

G = ∐_{w∈W} BwB.

(More accurately, we should write Bn_{w}B where n_{w} is a representative of w ∈ W in N. But, since any two representatives of w lie in the same coset of T ⊆ B, the double coset Bn_{w}B does not depend on the choice of the representative.)

Furthermore, the proof of the simplicity of the Chevalley groups and their twisted analogues is most economically performed using the simplicity criterion for abstract groups with a BN-pair in Bourbaki [25, Chap. IV, §2, no. 7].

Groups with a BN-pair play an important rôle in finite group theory. In fact, it is known that every finite simple group possesses a BN-pair, except for the cyclic groups of prime order, the alternating groups of degree ⩾ 5, and the 26 sporadic simple groups; see Gorenstein et al. [96]. Given a finite group G with a BN-pair, the irreducible factors of the Weyl group W are of type A_{n}, B_{n}, D_{n}, G_{2}, F_{4}, E_{6}, E_{7}, E_{8} or I_{2}(8). This follows from the classification by Tits [180] (rank ⩾ 3), Hering, Kantor, Seitz [98], [117] (rank 1) and Fong–Seitz [81] (rank 2). Note that there is only one case where W is not crystallographic: this is the case where W has a component of type I_{2}(8) (the dihedral group of order 16), which corresponds to the twisted groups of type F_{4} discovered by Ree (see Carter [42] or Steinberg [175]).

In another direction, BN-pairs with infinite Weyl groups arise naturally in the theory of p-adic groups; see Iwahori and Matsumoto [109].

2.17. Connected reductive algebraic groups. Here, we assume that the reader has some familiarity with the theory of linear algebraic groups; see Borel [23], Humphreys [106] or Springer [170]. Let G be a connected reductive algebraic group over an algebraically closed field K. Let B ⊆ G be a Borel subgroup. Then we have a semidirect product decomposition B = U T where U is the unipotent radical of B and T is a maximal torus. Let N = N_{G}(T), the normaliser of T in G. Then the groups B, N form a BN-pair in G; furthermore, W must be a finite Weyl group (and not just a Coxeter group as for general groups with a BN-pair). This is a deep, important result whose proof goes back to Chevalley [47]; detailed expositions can be found in the monographs by Borel [23, Chap. IV, 14.15], Humphreys [106, §29.1] or Springer [170, Chap. 8].

For example, in G = GL_{n}(K), the subgroup B of all upper triangular matrices is a Borel subgroup by the Lie–Kolchin Theorem (see, for example, Humphreys [106, 17.6]). Furthermore, we have a semidirect product decomposition B = U T where U ⊆ B is the normal subgroup consisting of all upper triangular matrices with 1 on the diagonal and T is the group of all diagonal matrices in B. Since K is infinite, it is easily checked that N = N_{G}(T), the group of all monomial matrices.

Returning to the general case, let us consider the Bruhat cells BwB (w ∈ W). These are locally closed subsets of G since they are orbits of B × B on G under left and right multiplication. The Zariski closure of BwB is given by

\overline{BwB} = ⋃_{y∈S(w)} ByB.

This yields the promised geometric description of the Bruhat–Chevalley order ⩽ on W (as defined in the remarks following Theorem 2.2.) The proof (see, for example, Springer [170, §8.5]) relies in an essential way on the fact that G/B is a projective variety.

3. Braid groups

3.1. The braid group of a complex reflection group. For a complex reflection group W ⩽ GL(V), V = ℂ^{n}, denote by A the set of its reflecting hyperplanes in V. The topological space

V^{reg} := V \ ⋃_{H∈A} H

is (pathwise) connected in its inherited complex topology. For a fixed base point x_{0} ∈ V^{reg} we define the pure braid group of W as the fundamental group P (W) := π_{1}(V^{reg}, x_{0}). Now W acts on V^{reg}, and by the Theorem of Steinberg (Theorem 1.6) the covering ¯ : V^{reg} → V^{reg}/W is Galois, with group W. This induces a short exact sequence

(1)  1 ⟶ P (W) ⟶ B(W) ⟶ W ⟶ 1

for the braid group B(W) := π_{1}(V^{reg}/W, x̄_{0}) of W.

If W = 𝔖_{n} in its natural permutation representation, the group B(W) is just the classical Artin braid group on n strings [4].

We next describe some natural generators of B(W). Let H ∈ A be a reflecting hyperplane. Let x_{H} ∈ H and r > 0 such that the open ball B(x_{H}, 2r) around x_{H} does not intersect any other reflecting hyperplane and x_{0} ∉ B(x_{H}, 2r). Choose a path γ : [0, 1] → V from the base point x_{0} to x_{H}, with γ(t) ∈ V^{reg} for t < 1. Let t_{0} be minimal subject to γ(t) ∈ B(x_{H}, r) for all t > t_{0}. Then γ′ := γ(t/t_{0}) is a path from x_{0} to γ(t_{0}). Then

λ : [0, 1] → B(x_{H}, 2r), t ↦ γ(t_{0}) exp(2πit/e_{H}),

where e_{H} = |W_{H}| is the order of the fixator of H in W, defines a closed path in the quotient V^{reg}/W. The homotopy class in B(W) of the composition γ′ ∘ λ ∘ γ^{′−1} is then called a braid reflection (see Broué [33]) or generator of the monodromy around H. Its image in W is a reflection s_{H} generating W_{H}, with non-trivial eigenvalue exp(2πi/e_{H}). It can be shown that B(W) is generated by all braid reflection, when H varies over the reflecting hyperplanes of W (Broué–Malle–Rouquier [38, Th. 2.17]).

Assume from now on that W is irreducible. Recall the definition of N, N^{∗} in Sections 1.2 and 1.7 as the number of reflections respectively of reflecting hyperplanes. The following can be shown without recourse to the classification of irreducible complex reflection groups:

3.2. Theorem (Bessis [11]). Let W ⩽ GL_{n}(ℂ) be an irreducible complex reflection group with braid group B(W). Let d be a degree of W which is a regular number for W and let r := (N + N^{∗})/d. Then r ∈ ℕ, and there exists a subset S = {s_{1}, …, s_{r}} ⊂ B(W) with:

(i) s_{1}, …, s_{r} are braid reflections, so their images s_{1}, …, s_{r} ∈ W are reflections.

(ii) S generates B(W), and hence S := {s_{1}, …, s_{r}} generates W.

(iii) There exists a finite set R of relations of the form w_{1} = w_{2}, where w_{1}, w_{2} are words of equal length in s_{1}, …, s_{r}, such that ⟨s_{1}, …, s_{r} | R⟩ is a presentation for B(W).

(iv) Let e_{s} denote the order of s ∈ S. Then ⟨s_{1}, …, s_{r} | R; s^{e_{s}} = 1 ∀s ∈ S⟩ is

a presentation for W, where now R is viewed as a set of relations on S.

(v) (s_{1} ⋯ s_{r})^{d} is central in B(W) and lies in P (W).

(vi) The product c := s_{1} ⋯ s_{r} is a ζ := exp(2πi/d)-regular element of W (hence has eigenvalues ζ^{−m_{1}}, …, ζ^{−m_{r}}).

It follows from the classification (see Table 1) that there always exists a regular degree. In many cases, for example if W is well-generated, the number (N +N^{∗})/r is regular, when r is chosen as the minimal number of generating reflections for W (so n ⩽ r ⩽ n + 1). Thus, in those cases B(W) is finitely presented on the same minimal number of generators as W. Under the assumptions (i) or (ii) of Theorem 1.16, the largest degree d_{n} is regular, whence Theorem 1.16 (iv) is a consequence of the previous theorem.

At present, presentations of the type described in Theorem 3.2 have been found for all but six irreducible types, by case-by-case considerations, see Bannai [5], Naruki [145], Broué–Malle–Rouquier [38]. For the remaining six groups, conjectural presentations have been found by Bessis and Michel using computer calculations.

For the case of real reflection groups, Brieskorn [28] and Deligne [61] determined the structure of B(W) by a nice geometric argument. They show that the generators in Theorem 3.2 (with r = n) can be taken as suitable preimages of the Coxeter generators, and the relations R as the Coxeter relations. For the case of W (A_{n}) = 𝔖_{n+1} of the classical braid group, this was first shown by Artin [4].

A topological space X is called K(π, 1) if all homotopy groups π_{i}(X) for i ≠ 1 vanish. The following is conjectured by Arnol’d to be true for all irreducible complex reflection groups:

3.3. Theorem. Assume that W is not of type G_{i}, i ∈ {24, 27, 29, 31, 33, 34}. Then V^{reg} and V^{reg}/W are K(π, 1)-spaces.

This was proved by a general argument for Coxeter groups by Deligne [61], after Fox and Neuwirth [82] showed it for type A_{n} and Brieskorn [29] for those of type different from H_{3}, H_{4}, E_{6}, E_{7}, E_{8}. For the non-real Shephard groups (non-real groups with Coxeter braid diagrams), it was proved by Orlik and Solomon [157]. The case of the infinite series G(de, e, r) has been solved by Nakamura [144]. In that case, there exists a locally trivial fibration

V^{reg}(G(de, e, n)) ⟶ V^{reg}(G(de, e, n − 1)),

with fiber isomorphic to ℂ minus m(de, e, n) points, where

m(de, e, n) := \{ \begin{matrix}(n − 1)de + 1 & for d ≠ 1, \\ (n − 1)(e − 1) & for d = 1. \end{matrix}

This induces a split exact sequence

1 ⟶ F_{m} ⟶ P (G(de, e, n)) ⟶ P (G(de, 1, n − 1)) ⟶ 1

for the pure braid group, with a free group F_{m} of rank m = m(de, e, n). In particular, the pure braid group has the structure of an iterated semidirect product of free groups (see Broué–Malle–Rouquier [38, Prop. 3.37]). 3.4. The center and regular elements. Denote by π the class in P (W) of the loop

[0, 1] → V^{reg}, t ↦ x_{0} exp(2πit).

Then π lies in the center Z(P (W)) of the pure braid group. Furthermore,

[0, 1] → V^{reg}, t ↦ x_{0} exp(2πit/|Z(W)|),

defines a closed path in V^{reg}/W, so an element β of B(W), which is again central. Clearly π = β^{|Z(W)|}.

The following was shown independently by Brieskorn–Saito [30] and Deligne [61] for Coxeter groups, and by Broué–Malle–Rouquier [38, Th. 2.24] for the other groups:

3.5. Theorem. Assume that W is not of type G_{i}, i ∈ {24, 27, 29, 31, 33, 34}. Then the center of B(W) is infinite cyclic generated by β, the center of P (W) is infinite cyclic generated by π, and the exact sequence (1) induces an exact sequence

1 ⟶ Z(P (W)) ⟶ Z(B(W)) ⟶ Z(W) ⟶ 1.

In their papers, Brieskorn–Saito [30] and Deligne [61] also solve the word problem and the conjugation problem for braid groups attached to real reflection groups.

For each H ∈ A choose a linear form α_{H} : V → ℂ with kernel H. Let e_{H} := |W_{H}|, the order of the minimal parabolic subgroup fixing H. The discriminant of W, defined as

δ := δ(W) := ∏_{H∈A} α_{H}^{e_{H}} ,

is then a W -invariant element of the symmetric algebra S(V^{∗}) of V^{∗}, well-defined up to non-zero scalars (Cohen [50, 1.8]). It thus induces a continuous function δ : V^{reg}/W → ℂ^{×}, hence by functoriality a group homomorphism π_{1}(δ) : B(W) → π_{1}(ℂ^{×}, 1) ≅ ℤ. For b ∈ B(W) let l(b) := π_{1}(δ)(b) denote the length of b. For example, every braid reflection s has length l(s) = 1, and we have

l(β) = (N + N^{∗})/|Z(W)| and hence l(π) = N + N^{∗}

by Broué–Malle–Rouquier [38, Cor. 2.21].

The elements b ∈ B(W) with l(b) ⩾ 0 form the braid monoid B^{+}(W). A dth root of π is by definition an element w ∈ B^{+} with w^{d} = π.

Let d be a regular number for W, and w a dth root of π. Assume that the image w of w in W is ζ-regular for some dth root of unity ζ (in the sense of 1.9). (This is for example the case if W is a Coxeter group by Broué-Michel [39, Thm. 3.12].) By Theorem 1.11(ii) the centraliser W (w) := C_{W} (w) is a reflection group on V (w, ζ), with reflecting hyperplanes the intersections of V (w, ζ) with the hyperplanes in A by Theorem 1.12(i). Thus the hyperplane complement of W (w) on V (w, ζ) is just V^{reg}(w) := V^{reg} ∩ V (w, ζ). Assuming that the base point x_{0} has been chosen in V^{reg}(w), this defines natural maps

P (W (w)) → P (W)  and  ψ_{w} : B(W (w)) → B(W).

By Broué–Michel [39, 3.4] the image of B(W (w)) in B(W) centralises w. It is conjectured (see Bessis–Digne–Michel [12, Conj. 0.1]) that ψ_{w} defines an isomorphism B(W (w)) ≅ C_{B(W)}(w). The following partial answer is known:

3.6. Theorem (Bessis–Digne–Michel [12, Th. 0.2]). Let W be an irreducible reflection group of type 𝔖_{n}, G(d, 1, n) or G_{i}, i ∈ {4, 5, 8, 10, 16, 18, 25, 26, 32}, and let w ∈ W be regular. Then ψ_{w} induces an isomorphism B(W (w)) ≅ C_{B(W)}(w)

This has also been proved by Michel [141, Cor. 4.4] in the case that W is a Coxeter group and w acts on W by a diagram automorphism. The injectivity of ψ_{w} was shown for all but finitely many types of W by Bessis [10, Th. 1.3].

The origin of Artin’s work on the braid group associated with the symmetric group lies in the theory of knots and links. We shall now briefly discuss this connection and explain the construction of the “HOMFLY–PT” invariant of knots and links (which includes the famous Jones polynomial as a special case). We follow the exposition in Geck–Pfeiffer [95, §4.5]. 3.7. Knots and links, Alexander and Markov theorem. If n is a positive integer, an oriented n-link is an embedding of n copies of the interval [0, 1] ⊂ ℝ into ℝ^{3} such that 0 and 1 are mapped to the same point (the orientation is induced by the natural ordering of [0, 1]); a 1-link is also called a knot. We are only interested in knots and links modulo isotopy, i.e., homeomorphic transformations which preserve the orientation. We refer to Birman [16], Crowell–Fox [57] or Burde–Zieschang [40] for precise versions of the above definitions.

By Artin’s classical interpretation of B(𝔖_{n}) as the braid group on n strings, each generator of B(𝔖_{n}) can be represented by oriented diagrams as indicated below; writing any g ∈ B(𝔖_{n}) as a product of the generators and their inverses, we also obtain a diagram for g, by concatenating the diagrams for the generators. “Closing” such a diagram by joining the end points, we obtain the plane projection of an oriented link in ℝ^{3}:

[figure: ^{s{−1 \atop i}}; …; …; 1; 2; i; i+1; n−1; n; 1; 2; i; i+1; n−1; n; ^{s^{i}}; …; …; 1; 2; i; i+1; n−1; n; 1; 2; i; i+1; n−1; n; g=s_{1}^{3}∈B(𝔖_{2}); ^{s^{1}}; ^{s^{1}}; ^{s^{1}}; 1; 2; 1; 2; closure of g]

By Alexander’s theorem (see Birman [16] or, for a more recent proof, Vogel [185]), every oriented link in ℝ^{3} is isotopic to the closure of an element in B(𝔖_{n}), for some n ⩾ 1. The question of when two links in ℝ^{3} are isotopic can also be expressed algebraically. For this purpose, we consider the infinite disjoint union

B_{∞} := ∐_{n⩾1} B(𝔖_{n}).

Given g, g′ ∈ B_{∞}, we write g ∼ g′ if one of the following relations is satisfied: (I) We have g, g′ ∈ B(𝔖_{n}) and g′ = x^{−1}gx for some x ∈ B(𝔖_{n}). (II) We have g ∈ B(𝔖_{n}), g′ ∈ B(𝔖_{n+1}) and g′ = gs_{n} or g′ = gs_{n}^{−1}.

The above two relations are called Markov relations. By a classical result due to Markov (see Birman [16] or, for a more recent proof, Traczyk [182]), two elements of B_{∞} are equivalent under the equivalence relation generated by ∼ if and only if the corresponding links obtained by closure are isotopic. Thus, to define an invariant of oriented links is the same as to define a map on B_{∞} which takes equal values on elements g, g′ ∈ B_{∞} satisfying (I) or (II).

We now consider the Iwahori–Hecke algebra H_{ℂ}(𝔖_{n}) of the symmetric group 𝔖_{n} over ℂ. By definition, H_{ℂ}(𝔖_{n}) is a quotient of the group algebra of B(𝔖_{n}), where we factor by an ideal generated by certain quadratic relations depending on two parameters u, v ∈ ℂ. This is done such that

T_{s_{i}}^{2} = uT_{1} + vT_{s_{i}} for 1 ⩽ i ⩽ n − 1,

where T_{s_{i}} denotes the image of the generator s_{i} of B(𝔖_{n}) and T_{1} denotes the identity element. For each w ∈ 𝔖_{n}, we have a well-defined element T_{w} such that

T_{w}T_{w′} = T_{ww′} whenever l(ww′) = l(w) + l(w′).

This follows easily from Matsumoto’s theorem 2.2. In fact, one can show that the elements {T_{w} | w ∈ 𝔖_{n}} form a ℂ-basis of H_{ℂ}(𝔖_{n}). (For more details, see the chapter on Hecke algebras.) The map w ↦ T_{w} (w ∈ 𝔖_{n}) extends to a well-defined algebra homomorphism from the group algebra of B(𝔖_{n}) over ℂ onto H_{ℂ}(𝔖_{n}). Furthermore, the inclusion 𝔖_{n−1} ⊆ 𝔖_{n} also defines an inclusion of algebras H_{ℂ}(𝔖_{n−1}) ⊆ H_{ℂ}(𝔖_{n}).

3.8. Theorem (Jones, Ocneanu [112]). There is a unique family of ℂ-linear maps τ_{n}: H_{ℂ}(𝔖_{n}) → ℂ (n ⩾ 1) such that the following conditions hold: (M1) τ_{1}(T_{1}) = 1; (M2) τ_{n+1}(hT_{s_{n}}) = τ_{n+1}(hT_{s_{n}}^{−1}) = τ_{n}(h) for all n ⩾ 1 and h ∈ H_{ℂ}(𝔖_{n}); (M3) τ_{n}(hh′) = τ_{n}(h′h) for all n ⩾ 1 and h, h′ ∈ H_{ℂ}(𝔖_{n}). Moreover, we have τ_{n+1}(h) = v^{−1}(1 − u)τ_{n}(h) for all n ⩾ 1 and h ∈ H_{ℂ}(𝔖_{n}).

In [112], Jones works with an Iwahori–Hecke algebra of 𝔖_{n} where the parameters are related by v = u − 1. The different formulation above follows a suggestion by J. Michel. It results in a simplification of the construction of the link invariants below. (The simplification arises from the fact that, due to the presence of two different parameters in the quadratic relations, the “singularities” mentioned in [112, p. 349, Notes (1)] simply disappear.) Generalisations of Theorem 3.8 to types G(d, 1, n) and D_{n} have been found in Geck–Lambropoulou [92], Lambropoulou [125] and Geck [87]. 3.9. The HOMFLY-PT polynomial. We can now construct a two-variable invariant of oriented knots and links as follows. Consider an oriented link L and assume that it is isotopic to the closure of g ∈ B(𝔖_{n}) for n ⩾ 1. Then we set

X_{L}(u, v) := τ_{n}(ḡ) ∈ ℂ with τ_{n} as in Theorem 3.8.

Here, ḡ denotes the image of g under the natural map ℂ[B(𝔖_{n})] → H_{ℂ}(𝔖_{n}), w ↦ T_{w} (w ∈ 𝔖_{n}). It is easily checked that X_{L}(u, v) can be expressed as a Laurent polynomial in u and v; the properties (M2) and (M3) make sure that τ_{n}(ḡ) does not depend on the choice of g. If we make the change of variables u = t^{2} and v = tx, we can identify the above invariant with the HOMFLY-PT polynomial P_{L}(t, x) discovered by Freyd et al. [83] and Przytycki–Traczyk [160]; see also Jones [112, (6.2)]. Furthermore, the Jones polynomial J_{L}(t) is obtained by setting u = t^{2}, v = \sqrt{t}(t − 1) (see [112, §11]). Finally, setting u = 1 and v = \sqrt{t} − 1/\sqrt{t}, we obtain the classical Alexander polynomial A_{L}(t) whose definition can be found in Crowell– Fox [57].

For a survey about recent developments in the theory of knots and links, especially since the discovery of the Jones polynomial, see Birman [17]. 3.10. Further aspects of braid groups. One of the old problems concerning braid groups is the question whether or not they are linear, i.e., whether there exists a faithful linear representation on a finite dimensional vector space. Significant progress has been made recently on this problem. Krammer [122] and Bigelow [15] proved that the classical Artin braid group is linear. Then Digne [69] and Cohen– Wales [53] extended this result and showed that all Artin groups of crystallographic type have a faithful representation of dimension equal to the number of reflections of the associated Coxeter group.

On the other hand, there is one particular representation of the braid group associated with 𝔖_{n}, the so-called Burau representation (see Birman [16], for which it has been a long-standing problem to determine for which values of n it is faithful. Moody [142] showed that it is not faithful for n ⩾ 10; this bound was improved by Long and Paton [132] to 6. Recently, Bigelow [14] showed that the Burau representation is not faithful already for n = 5. (It is an old result of Magnus and Peluso that the Burau representation is faithful for n = 3.)

In a different direction, Deligne’s and Brieskorn–Saito’s solution of the word and conjugacy problem in braid groups led to new developments in combinatorial group and monoid theory; see, for example, Dehornoy and Paris [60] and Dehornoy [59].

4. Representation theory

In this section we report about the representation theory of finite complex reflection groups. 4.1. Fields of definition. Let W be a finite complex reflection group on V. Let K_{W} denote the character field of the reflection representation of W, that is, the field generated by the traces tr_{V} (w), w ∈ W. It is easy to see that the reflection representation can be realised over K_{W} (see for example [7, Prop. 7.1.1]). But we have a much stronger statement:

4.2. Theorem (Benard [6], Bessis [9]). Let W be a complex reflection group. Then the field K_{W} is a splitting field for W.

The only known proof for this result is case-by-case, treating the reflection groups according to the Shephard–Todd classification.

The field K_{W} has a nice description at least in the case of well-generated groups, that is, irreducible groups generated by dim(V) reflections. In this case, the largest degree d_{n} of W is regular, so there exists an element c := s_{1} ⋯ s_{n} as in the Theorem 3.2(vi) of Bessis, called a Coxeter element of W, with eigenvalues ζ^{−m_{1}}, …, ζ^{−m_{n}} in the reflection representation, where ζ := exp(2πi/d_{n}) and the m_{i} are the exponents of W.

If W is a real reflection group, then W is a Coxeter group associated with some Coxeter matrix M (see Theorem 2.4) and we have

K_{W} = ℚ(cos(2π/m_{st}) | s, t ∈ S) ⊂ ℝ.

In particular, this shows that K_{W} = ℚ if W is a finite Weyl group.

For well-generated irreducible complex reflection groups W ⩽ GL(V), the field of definition K_{W} is generated over ℚ by the coefficients of the characteristic polynomial on V of a Coxeter element, see Malle [139, Th. 7.1]. This characterisation is no longer true for non-well generated reflection groups. 4.3. Macdonald–Lusztig–Spaltenstein induction. Let W be a complex reflection group on V. Recall from Section 1.7 the definition of the fake degree R_{χ} of an irreducible character χ ∈ Irr(W). The b-invariant b_{χ} of χ is defined as the order of vanishing of R_{χ} at x = 0, that is, as the minimum of the exponents e_{i}(χ) of χ. The coefficient of x^{b_{χ}} in R_{χ} is denoted by γ_{χ}.

4.4. Theorem (Macdonald [135], Lusztig–Spaltenstein [134]). Let W be a complex reflection group on V, W′ a reflection subgroup (on V_{1} := V/C_{V} (W′)). Let ψ be an irreducible character of W′ such that γ_{ψ} = 1. Then Ind_{W′}^{W}(ψ) has a unique irreducible constituent χ ∈ Irr(W) with b_{χ} = b_{ψ}. This satisfies γ_{χ} = 1. All other constituents have b-invariant bigger than b_{ψ}

The character χ ∈ Irr(W) in Theorem 4.4 is called the j-induction j_{W′}^{W}(ψ) of the character ψ ∈ Irr(W′). Clearly, j-induction is transitive; it is also compatible with direct products.

An important example of characters ψ with γ_{ψ} = 1 is given by the determinant character det_{V} : W → ℂ^{×} of a complex reflection group (see Geck–Pfeiffer [95, Th. 5.2.10]). 4.5. Irreducible characters. There is no general construction of all irreducible representations of a complex reflection group known. Still, we have the following partial result:

4.6. Theorem (Steinberg). Let W ⩽ GL(V) be an irreducible complex reflection group. Then the exterior powers Λ^{i}(V), 1 ⩽ i ⩽ dim V, are irreducible, pairwise non-equivalent representations of W.

A proof in the case of well-generated groups can be found in Bourbaki [25, V, §2, Ex. 3(d)] and Kane [114, Thm. 24-3 A], for example. The general case then follows with Corollary 1.15.

We now give some information on the characters of individual reflection groups. The irreducible characters of the symmetric group 𝔖_{n} were determined by Frobenius [84], see also Macdonald [136] and Fulton [85]. Here we follow the exposition in Geck–Pfeiffer [95, 5.4].

Let λ = (λ_{1}, …, λ_{r}) ⊢ n be a partition of n. The corresponding Young subgroup 𝔖_{λ} of 𝔖_{n} is the common setwise stabiliser {1, …, λ_{1}}, {λ_{1} + 1, …, λ_{1} + λ_{2}}, …, abstractly isomorphic to 𝔖_{λ} = 𝔖_{λ_{1}} × … × 𝔖_{λ_{r}}. This is a parabolic subgroup of 𝔖_{n} in the sense of Section 1.5. For any m ⩾ 1, let 1_{m}, ϵ_{m} denote the trivial respectively the sign character of 𝔖_{m}. For each partition we have the two induced characters

π_{λ} := Ind_{𝔖_{λ}}^{𝔖_{n}}(1_{λ_{1}}# … #1_{λ_{r}}), θ_{λ} := Ind_{𝔖_{λ}}^{𝔖_{n}}(ϵ_{λ_{1}}# … #ϵ_{λ_{r}}).

Then π_{λ} and θ_{λ^{∗}} have a unique irreducible constituent χ_{λ} ∈ Irr(𝔖_{n}) in common, where λ^{∗} denotes the partition dual to λ. This constituent can also be characterised in terms of j-induction as

χ_{λ} = j_{𝔖_{λ}}^{𝔖_{n}}(1_{λ_{1}}# … #1_{λ_{r}}).

Then the χ_{λ} are mutually distinct and exhaust the irreducible characters of 𝔖_{n}, so Irr(𝔖_{n}) = {χ_{λ} | λ ⊢ n}. From the above construction it is easy to see that all χ_{λ} are afforded by rational representations.

The construction of the irreducible characters of the imprimitive group G(d, 1, n) goes back at least to Osima [158] (see also Read [163], Hughes [102], Bessis [9]) via their abstract structure as wreath product C_{d} ≀ 𝔖_{n}. Let us fix d ⩾ 2 and write W_{n} := G(d, 1, n). A d-tuple α = (α_{0}, α_{1}, …, α_{d−1}) of partitions α_{i} ⊢ n_{i} with ∑ n_{i} = n is called a d-partition of n. We denote by W_{α} the natural subgroup W_{n_{0}} × … × W_{n_{d−1}} of W_{n}, where α_{j} ⊢ n_{j}, corresponding to the Young subgroup 𝔖_{n_{0}} × … × 𝔖_{n_{d−1}} of 𝔖_{n}. Via the natural projection W_{n_{j}} → 𝔖_{n_{j}} the characters of 𝔖_{n_{j}} may be regarded as characters of W_{n_{j}}. Thus each α_{j} defines an irreducible character χ_{α_{j}} of W_{n_{j}}. For any m, let ζ_{d} : W_{m} → ℂ^{×} be the linear character defined by ζ_{d}(t_{1}) = exp(2πi/d), ζ_{d}(t_{i}) = 1 for i > 1, with the standard generators t_{i} from 1.13. Then for any d-partition α of n we can define a character χ_{α} of W_{n} as the induction of the exterior product

χ_{α} := Ind_{W_{α}}^{W_{n}} (χ_{α_{0}}#(χ_{α_{1}} ⊗ ζ_{d})# … #(χ_{α_{d−1}} ⊗ ζ_{d}^{d−1})) .

By Clifford-theory χ_{α} is irreducible, χ_{α} ≠ χ_{β} if α ≠ β, and all irreducible characters of W_{n} arise in this way, so

Irr(W_{n}) = {χ_{α} | α = (α_{0}, …, α_{d−1})⊢_{d}n}.

We describe the irreducible characters of G(de, e, n) in terms of those of W_{n} := G(de, 1, n). Recall that the imprimitive reflection group G(de, e, n) is generated by the reflections t_{2}, t̃_{2} := t_{1}^{−1}t_{2}t_{1}, t_{3}, …, t_{n}, and t_{1}^{e}. Denote by π the cyclic shift on de-partitions of n, i.e.,

π(α_{0}, …, α_{de−1}) = (α_{1}, …, α_{de−1}, α_{0}) .

By definition we then have χ_{π(α)} ⊗ ζ_{de} = χ_{α}. Let s_{e}(α) denote the order of the stabiliser of α in the cyclic group ⟨π^{d}⟩. Then upon restriction to G(de, e, n) the irreducible character χ_{α} of W_{n} splits into s_{e}(α) different irreducible constituents, and this exhausts the set of irreducible characters of G(de, e, n). More precisely, let α be a de-partition of n with ẽ:= s_{e}(α), W_{α,e} := W_{α} ∩ G(de, e, n), and ψ_{α} the restriction of

χ_{α_{0}}#(χ_{α_{1}} ⊗ ζ_{de})# … #(χ_{α_{de−1}} ⊗ ζ_{de}^{de−1})

to W_{α,e}. Then ψ_{α} is invariant under the element σ := (t_{2} ⋯ t_{n})^{n/ẽ} (note that ẽ = s_{e}(α) divides n), and it extends to the semidirect product W_{α,e}.⟨σ⟩. The different extensions of ψ_{α} induced to G(de, e, n) then exhaust the irreducible constituents of the restriction of χ_{α} to G(de, e, n). Thus, we may parametrise Irr(G(de, e, n)) by de-partitions of n up to cyclic shift by π^{d} in such a way that any α stands for s_{e}(α) different characters.

In order to describe the values of the irreducible characters we need the following definitions. We identify partitions with their Young diagrams. A d-partition α is called a hook if it has just one non-empty part, which is a hook (i.e., does not contain a 2 × 2-block). The position of the non-empty part is then denoted by τ(α). If α, β are d-partitions such that β_{i} is contained in α_{i} for all 0 ⩽ i ⩽ d − 1, then α \ β denotes the d-partition (α_{i} \ β_{i} | 0 ⩽ i ⩽ d − 1), where α_{i} \ β_{i} is the set theoretic difference of α_{i} and β_{i}. If α \ β is a hook, we denote by l_{β}^{α} the number of rows of the hook (α \ β)_{τ(α\β)} minus 1. With these notations the values of the irreducible characters of G(d, 1, n) can be computed recursively with a generalised Murnaghan–Nakayama rule (see also Stembridge [178]):

4.7. Theorem (Osima [158]). Let α and γ be d-partitions of n, let m be a part of γ_{t} for some 1 ⩽ t ⩽ d, and denote by γ′ the d-partition of n − m obtained from γ by deleting the part m from γ_{t}. Then the value of the irreducible character χ_{α} on an element of G(d, 1, n) with cycle structure γ is given by

χ_{α}(γ) = ∑_{α\β⊢_{d}m} ζ_{d}^{st}(−1)^{l_{β}^{α}}χ_{β}(γ′)

where the sum ranges over all d-partitions β of n − m such that α \ β is a hook and where s = τ(α \ β).

An overview of the irreducible characters of the exceptional groups (and of their projective characters) is given in Humphreys [107]. All character tables of irreducible complex reflection groups are also available in the computer algebra system CHEVIE [89].

4.8. Fake degrees. The fake degrees (introduced in Section 1.7) of all complex reflection groups are known. For the symmetric groups, they were first determined by Steinberg [173], as generic degrees of the unipotent characters of the general linear groups over a finite field. From that result, the fake degrees of arbitrary imprimitive reflection groups can easily be derived (see Malle [137, Bem. 2.10 and 5.6]).

4.9. Theorem (Steinberg [173], Lusztig [133]). The fake degrees of the irreducible complex reflection groups G(de, e, n) are given as follows:

(i) Let χ ∈ Irr(G(d, 1, n)) be parameterised by the d-partition (α_{0}, …, α_{d−1}), where α_{i} = (α_{i1} ⩾ … ⩾ α_{im_{i}}) ⊢ n_{i}, and let (S_{0}, …, S_{d−1}), where S_{i} = (α_{i1}+m_{i}−1, …, α_{im_{i}}), denote the corresponding tuple of β-numbers. Then

R_{χ} = ∏_{i=1}^{n}(x^{id} − 1) ∏_{i=0}^{d−1} \frac{Δ(S_{i}, x^{d})x^{in_{i}}}{Θ(S_{i}, x^{d})x^{d\binom{m_{i}−1}{2}+d\binom{m_{i}−2}{2}+...}} ,

where, for a finite subset S ⊂ ℕ,

Δ(S, x) := ∏_{{λ,λ′∈S \atop λ′<λ}} (x^{λ} − x^{λ′}), Θ(S, x) := ∏_{λ∈S} ∏_{h=1}^{λ}(x^{h} − 1).

(ii) The fake degree of χ ∈ Irr(G(de, e, n)) is obtained from the fake degrees in G(de, 1, n) as

R_{χ} = \frac{x^{nd} − 1}{x^{nde} − 1} ∑_{ψ∈Irr(G(de,1,n))} ⟨χ, ψ|_{G(de,e,n)}⟩ R_{ψ}.

For exceptional complex reflection groups, the fake degrees can easily be computed, for example in the computer algebra system CHEVIE [89]. In the case of exceptional Weyl groups, they were first studied by Beynon–Lusztig [13].

The fake degrees of reflection groups satisfy a remarkable palindromicity property:

4.10. Theorem (Opdam [150, 151], Malle [139]). Let W be a complex reflection group. There exists a permutation δ on Irr(W) such that for every χ ∈ Irr(W) we have

R_{χ}(x) = x^{c}R_{δ(χ̄)}(x^{−1}),

where c = ∑_{r}(1 − χ(r)/χ(1)), the sum running over all reflections r ∈ W.

The interest of this result also lies in the fact that the permutation δ is strongly related to the irrationalities of characters of the associated Hecke algebra. Theorem 4.10 was first observed empirically by Beynon–Lusztig [13] in the case of Weyl groups. Here δ is non-trivial only for characters such that the corresponding character of the associated Iwahori-Hecke algebra is non-rational. An a priori proof in this case was later given by Opdam [150]. If W is complex, Theorem 4.10 was verified by Malle [139, Thm. 6.5] in a case-by-case analysis. Again in all but possibly finitely many cases, δ comes from the irrationalities of characters of the associated Hecke algebra. Opdam [151, Thm. 4.2 and Cor. 6.8] gives a general argument which proves Theorem 4.10 under a suitable assumption on the braid group B(W).

The above discussion is exclusively concerned with representations over a field of characteristic 0 (the “semisimple case”). We close this chapter with some remarks concerning the modular case. 4.11. Modular representations of 𝔖_{n}. Frobenius’ theory (as developed further by Specht, James and others) yields a parametrisation of Irr(𝔖_{n}) and explicit formulas for the degrees and the values of all irreducible characters. As soon as we consider representations over a field of characteristic p > 0, the situation changes drastically. James [110] showed that the irreducible representations of 𝔖_{n} still have a natural parametrisation, by so-called p-regular partitions. Furthermore, the decomposition matrix relating representations in characteristic 0 and in characteristic p has a lower triangular shape with 1s on the diagonal. This result shows that, in principle, a knowledge of the irreducible representations of 𝔖_{n} in characteristic p is equivalent to the knowledge of the decomposition matrix.

There are a number of results known about certain entries of that decomposition matrix, but a general solution to this problem is completely open; see James [110] for a survey. Via the classical Schur algebras (see Green [97]) it is known that the decomposition numbers of 𝔖_{n} in characteristic p can be obtained from those of the finite general linear group GL_{n}(𝔽_{q}) (where q is a power of p). Now, at first sight, the problem of computing the decomposition numbers for GL_{n}(𝔽_{q}) seems to be much harder than for 𝔖_{n}. However, Erdmann [77] has shown that, if one knows the decomposition numbers of 𝔖_{m} (for sufficiently large values of m), then one will also know the decomposition numbers of GL_{n}(𝔽_{q}). Thus, the problem of determining the decomposition numbers for symmetric groups appears to be as difficult as the corresponding problem for general linear groups.

In a completely different direction, Dipper and James [70] showed that the decomposition numbers of 𝔖_{n} can also be obtained from the so-called q-Schur algebra, which is defined in terms of the Hecke algebra of 𝔖_{n}. Thus, the problem of determining the representations of 𝔖_{n} in characteristic p is seen to be a special case of the more general problem of studying the representations of Hecke algebras associated with finite Coxeter groups. This is discussed in more detail in the chapter on Hecke algebras. In this context, we just mention here that James’ result on the triangularity of the decomposition matrix of 𝔖_{n} is generalised to all finite Weyl groups in Geck [88].

5. Hints for further reading

Here, we give some hints on topics which were not touched in the previous sections. 5.1. Crystallographic reflection groups. Let W be a discrete subgroup of the group of all affine transformations of a finite-dimensional affine space E over K = ℂ or K = ℝ, generated by affine reflections. If W is finite, it necessarily fixes a point and W is a finite complex reflection group. If W is infinite and K = ℝ, the irreducible examples are precisely the affine Weyl groups (see Section 2.14). In the complex case K = ℂ there are two essentially different cases. If E/W is compact, the group W is called crystallographic. The noncrystallographic groups are now just the complexifications of affine Weyl groups. The crystallographic reflection groups have been classified by Popov [159] (see also Kaneko–Tokunaga–Yoshida [181, 115] for related results). As in the real case they are extensions of a finite complex reflection group W_{0} by an invariant lattice which is generated by roots for W_{0}.

It turns out that presentations for these groups can be obtained as in the case of affine Weyl groups by adding a further generating reflection corresponding to a highest root in a root system for W_{0} (see Malle [138]). As for the finite complex reflection groups in Section 3.1, the braid group B(W) of W is defined as the fundamental group of the hyperplane complement. For many of the irreducible crystallographic complex reflection groups it is known that a presentation for B(W) can be obtained by ommitting the order relations from the presentation of W described above (see Dũng [72] for affine groups, Malle [138] for the complex case).

5.2. Quaternionic reflection groups. Cohen [51] has obtained the classification of finite reflection groups over the quaternions. This is closely related to finite linear groups over ℂ generated by bireflections, that is, elements of finite order which fix pointwise a subspace of codimension 2. Indeed, using the identification of the quaternions as a certain ring of 2 × 2-matrices over the complex numbers, reflections over the quaternions become complex bireflections. The primitive bireflection groups had been classified previously by Huffman–Wales [101, 188]. One of the examples is a 3-dimensional quaternionic representation of the double cover of the sporadic Hall–Janko group J_{2}, see Wilson [190].

Presentations for these groups resembling the Coxeter presentations for Weyl groups are given by Cohen [52].

In recent work on the McKay correspondence, quaternionic reflection groups play an important rôle under the name of symplectic reflection groups in the construction of so-called symplectic reflection algebras, see for example Etingof–Ginzburg [78].

5.3. Reflection groups over finite fields. Many of the general results for complex reflection groups presented in Section 1 are no longer true for reflection groups over fields of positive characteristic. Most importantly, the ring of invariants of such a reflection group is not necessarily a polynomial ring. Nevertheless we have the following criterion due to Serre [25, V.6, Ex. 8] and Nakajima [143], generalising Theorem 1.6:

5.4. Theorem (Serre, Nakajima [143]). Let V be a finite-dimensional vector space over a field K and W ⩽ GL(V) a finite group such that K[V]^{W} is polynomial. Then the point-wise stabiliser of any subspace U ⩽ V has polynomial ring of invariants (and thus is generated by reflections).

The irreducible reflection groups over finite fields were classified by Wagner [186, 187] and Zalesskiĭ–Serežkin [191], the determination of transvection groups was completed by Kantor [116]. In addition to the modular reductions of complex reflection groups, there arise the infinite families of classical linear, symplectic, unitary and orthogonal groups, as well as some further exceptional examples. For a complete list see for example Kemper–Malle [118, Section 1].

The results of Wagner [187] and Kantor [116] are actually somewhat stronger, giving a classification of all indecomposable reflection groups W over finite fields of characteristic p for which the maximal normal p-subgroup is contained in the intersection W′ ∩ Z(W) of the center with the derived group.

Using this classification, the irreducible reflection groups over finite fields with polynomial ring of invariants could be determined, leading to the following criterion:

5.5. Theorem (Kemper–Malle [118]). Let V be a finite-dimensional vector space over K, W ⩽ GL(V) a finite irreducible linear group. Then K[V]^{W} is a polynomial ring if and only if W is generated by reflections and the pointwise stabiliser in W of any nontrivial subspace of V has a polynomial ring of invariants.

The list of groups satisfying this criterion can be found in [118, Thm. 7.2]. That paper also contains some information on indecomposable groups.

It is an open question whether at least the field of invariants of a reflection group in positive characteristic is purely transcendental (by Kemper–Malle [119] the answer is positive in the irreducible case).

For further discussions of modular invariant theory of reflection groups see also Derksen–Kemper [68, 3.7.4].

5.6. p-adic reflection groups. Let R be an integral domain, L an R-lattice of finite rank, i.e., a torsion-free finitely generated R-module, and W a finite subgroup of GL(L) generated by reflections. Again one can ask under which conditions the invariants of W on the symmetric algebra R[L] of the dual L^{∗} are a graded polynomial ring. In the case of Weyl groups Demazure shows the following extension of Theorem 1.3:

5.7. Theorem (Demazure [63]). Let W be a Weyl group, L the root lattice of W, and R a ring in which all torsion primes of W are invertible. Then the invariants of W on R[L] are a graded polynomial algebra, and R[L] is a free graded module over R[L]^{W}.

In the case of general lattices for reflection groups, the following example may be instructive: Let W = 𝔖_{3} the symmetric group of degree 3. Then the weight lattice L of S_{3}, considered as ℤ_{3}-lattice, yields a faithful reflection representation of 𝔖_{3} with the following property: ℤ_{3}[L]^{𝔖_{3}} is not polynomial, while both the reflection representations over the quotient field ℚ_{3} and over the residue field 𝔽_{3} have polynomial invariants, the first with generators in degrees 2 and 3, the second with generators in degrees 1 and 6.

The list of all irreducible p-adic reflection groups, that is, reflection groups over the field of p-adic numbers ℚ_{p}, was obtained by Clark–Ewing [49], building on the Theorem of Shephard–Todd. We reproduce it in Table 6.

Using a case-by-case argument based on the Clark–Ewing classification and his own classification of p-adic lattices for reflection groups, Notbohm [149] was able to determine all finite reflection groups W over the ring of p-adic integers ℤ_{p},

Table 6. p-adic reflection groups

W  conditions
G(de, e, n) ((d, n) ≠ (1, 2)) p ≡ 1 (mod de); none when de = 2
G(e, e, 2) (e⩾3)  p ≡ ±1 (mod e); none when e = 3, 4, 6
𝔖_{n+1}  —
W conditions
G_{4} p ≡ 1 (mod 3)
G_{5} p ≡ 1 (mod 3)
G_{6} p ≡ 1 (mod 12)
G_{7} p ≡ 1 (mod 12)
G_{8} p ≡ 1 (mod 4)
G_{9} p ≡ 1 (mod 8)
G_{10} p ≡ 1 (mod 12)
G_{11} p ≡ 1 (mod 24)
G_{12} p ≡ 1, 3 (mod 8)
G_{13} p ≡ 1 (mod 8)
G_{14} p ≡ 1, 19 (mod 24)
G_{15} p ≡ 1 (mod 24)
G_{16} p ≡ 1 (mod 5)
G_{17} p ≡ 1 (mod 20)
G_{18} p ≡ 1 (mod 15)
G_{19} p ≡ 1 (mod 60)
G_{20} p ≡ 1, 4 (mod 15)

W conditions
G_{21} p ≡ 1, 49 (mod 60)
G_{22} p ≡ 1, 9 (mod 20)
G_{23} p ≡ 1, 4 (mod 5)
G_{24} p ≡ 1, 2, 4 (mod 7)
G_{25} p ≡ 1 (mod 3)
G_{26} p ≡ 1 (mod 3)
G_{27} p ≡ 1, 4 (mod 15)
G_{28} —
G_{29} p ≡ 1 (mod 4)
G_{30} p ≡ 1, 4 (mod 5)
G_{31} p ≡ 1 (mod 4)
G_{32} p ≡ 1 (mod 3)
G_{33} p ≡ 1 (mod 3)
G_{34} p ≡ 1 (mod 3)
G_{35} —
G_{36} —
G_{37} —

p > 2, with polynomial ring of invariants. This was subsequently extended by Andersen–Grodal–Møller–Viruel [3] to include the case p = 2.

5.8. p-compact groups. The p-adic reflection groups play an important rôle in the theory of so-called p-compact groups, which constitute a homotopy theoretic analogue of compact Lie groups. By definition, a p-compact group is a p-complete topological space BX such that the homology H_{∗}(X; 𝔽_{p}) of the loop space X = ΩBX is finite. Examples for p-compact groups are p-completions of classifying spaces of compact Lie groups. Further examples were constructed by Clark–Ewing [49], Aguadé [2], Dwyer–Wilkerson [73] and Notbohm [148]. To each p-compact group X Dwyer–Wilkerson [74] associate a maximal torus (unique up to conjugacy) together with a ‘Weyl group’, which comes equipped with a representation as a reflection group over the p-adic integers ℤ_{p}, which is faithful if X is connected. Conversely, by a theorem of Andersen et al. [3] a connected p-compact group, for p > 2, is determined up to isomorphism by its Weyl group data, that is, by its Weyl group in a reflection representation on a ℤ_{p}-lattice.

It has been shown that at least for p > 2 all p-adic reflection groups (as classified by Clark–Ewing) and all their ℤ_{p}-reflection representations arise in that way (see Andersen et al. [3], Notbohm [148] and also Adams–Wilkerson [1], Aguadé [2]).

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M.G.: Department of Mathematical Sciences, King’s College, Aberdeen University, Aberdeen AB24 3UE, Scotland, UK

E-mail address: geck@maths.abdn.ac.uk

G.M.: FB Mathematik, Universität Kaiserslautern, Postfach 3049, D – 67653 Kaiserslautern, Germany.

E-mail address: malle@mathematik.uni-kl.de