Weyl groups, affine Weyl groups and reflection groups

Raphaël Rouquier

Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 20 Clarkson Road, Cambridge, CB3 0EH, U.K. and UMR 9994 du CNRS, Université Paris 7–Denis Diderot, Case 7012, 2 Place Jussieu, F–75251 Paris Cedex 05, France

1 Introduction

This paper is a survey of some of the basic results pertaining to reflection groups.

In §2, we start with the basic concepts and properties of Coxeter groups, such as the Exchange Lemma and in §4 we construct the geometric representation. Sections 3 and 5 are devoted to finite real reflection groups and finite Coxeter groups and §6 concerns Weyl groups, which are crystallographic reflection groups. Weyl groups give rise to affine Weyl groups, studied in §7. The Iwahori-Hecke algebra of a Coxeter group is introduced in §8, after a discussion on braid groups. Finite complex reflection groups are the subject of §9, where we describe the infinite families. Finally, we explain in §10 how the topology of the hyperplane complement allows to define braid groups and Iwahori-Hecke algebras for finite complex reflection groups.

This paper is expository : most proofs are to be found in [Bki] or [Hu] for §2-8 and in [BrMaRo] for §9-10.

2 Coxeter groups

Let W be a group and S a set of (distinct) generators of W of order 2. For s, s′ ∈ S, we denote by m_{s,s′} ∈ {1, 2, … } ∪ {∞} the order of the product ss′.

Definition 2.1 The pair (W, S) is a Coxeter system if W has a presentation by generators and relations given by the set of generators S and the relations :

s^{2} = 1 for s ∈ S,

\underbrace{ss′ss′ ⋯}_{m_{ss′} terms} = \underbrace{s′ss′s ⋯}_{m_{ss′} terms} for those s, s′ ∈ S such that m_{s,s′} is finite.

We then say also that W is a Coxeter group. The relations ss′ss′ ⋯ = s′ss′s ⋯ are called braid relations. The rank of the system is the cardinality of S. The matrix of the Coxeter system (W, S) is (m_{s,s′})_{s,s′∈S} ; it has values in

{1, 2, … } ∪ {∞}.

This is a symmetric matrix with diagonal entries 1 and off-diagonal entries at least 2. A matrix with such properties is called a Coxeter matrix. We will see (Theorem 4.1) that every Coxeter matrix is the matrix of a Coxeter system (in a group given by generators and relations as in the definition, with (m_{s,s′}) an abstract Coxeter matrix, it isn’t obvious that ss′ will have order m_{s,s′}).

The graph associated with (W, S) is the graph with set of vertices S and edges {s, s′} when m_{s,s′} ≥ 3. Furthermore, the edge is then labelled by m_{s,s′}.

Some examples.

(i) The symmetric group 𝔖_{n} = 𝔖({1, 2, …, n}). Let s_{i} = (i, i + 1) and S_{𝔖_{n}} = {s_{1}, …, s_{n−1}}. Then, (𝔖_{n}, S_{𝔖_{n}}) is a Coxeter system (of type A_{n−1}) with graph

[figure: s_{1}; s_{2}; s_{3}; s_{n−2}; s_{n−1}] (the label is omitted when it is 3).

It has rank n − 1.

(ii) The hyperoctahedral group B_{n}, i.e., the group of n×n monomial matrices with non-zero entries in {±1}. It contains 𝔖_{n}, viewed as the group of permutation matrices, as a subgroup. Let s_{0} = diag(−1, 1, ..., 1) and S_{B_{n}} = {s_{0}, s_{1}, …, s_{n−1}}, with s_{i}, i ≥ 1 as in (i). Then, (B_{n}, S_{B_{n}}) is a Coxeter system of rank n with graph

[figure: s_{0}; s_{1}; s_{2}; s_{3}; s_{n−2}; s_{n−1}] (where [figure]stands for [figure: 4]).

(iii) The dihedral group I_{2}(m) : this is the symmetry group of a regular m-gon (i.e., the subgroup of the group of isometries of the plane fixing the m-gon), m ≥ 2.

[figure: H_{1}; H_{2}]

Let H_{1} be a line containing the center of the polygon and one of its vertices. Let H_{2} be a line containing the center of the polygon and such that the angle between H_{1} and H_{2} is π/m. I_{2}(m) is generated by the orthogonal reflections t_{1} and t_{2} with respect to H_{1} and H_{2}. (I_{2}(m), {t_{1}, t_{2}}) is a Coxeter system with graph

[figure: t_{1}; t_{2}; m](m ≥ 3) or [figure: t_{1}; t_{2}](m = 2).

The group I_{2}(m) has a decomposition I_{2}(m) = ⟨t_{1}t_{2}⟩ ⋊ ⟨t_{1}⟩. The subgroup ⟨t_{1}t_{2}⟩ is the subgroup of rotations, it has order m. The action of ⟨t_{1}⟩ ≃ {±1} on Z/mZ in this decomposition is given by multiplication.

This suggests a construction for m = ∞ : we denote by Ã_{1} the group Z ⋊ {±1}, where {±1} acts by multiplication on Z. Let t_{1} = (0, −1) and t_{2} = (−1, −1). Then, (Ã_{1}, {t_{1}, t_{2}}) is a Coxeter system with graph

[figure: t_{1}; t_{2}; ∞]

The dihedral groups are the groups I_{2}(m), 2 ≤ m < ∞ and Ã_{1}.

Note that every rank 2 Coxeter system is isomorphic to the Coxeter system of a dihedral group. In particular, the Coxeter systems for 𝔖_{3} and I_{2}(3) are isomorphic, as well as those for B_{2} and I_{2}(4).

The following theorem [Bki, Chap. IV, §1, Théorème 2] is an easy consequence of Theorem 4.1 below :

Theorem 2.2 Let (W, S) be a Coxeter system, S′ a subset of S and W′ the subgroup of W generated by S′. Then, (W′, S′) is a Coxeter system with Coxeter matrix the submatrix of the Coxeter matrix of (W, S) given by S′.

A Coxeter system is irreducible if its associated graph is connected. All systems in the previous examples are irreducible, except I_{2}(2).

If S is the disjoint union of two subsets S_{1} and S_{2} and no vertex of S_{1} is connected to a vertex of S_{2}, then W = W_{1} × W_{2}, where W_{i} is the subgroup of W generated by S_{i}.

Remark 1 Note that for m odd, I_{2}(2m) ≃ I_{2}(m) × 𝔖_{2}, but (I_{2}(2m), {s, s′}) is nevertheless irreducible for m > 1 !

Let w ∈ W. The length of w, l(w), is the smallest integer m such that w is the product of m elements of S.

A decomposition w = s_{1} ⋯ s_{m} with s_{1}, …, s_{m} ∈ S is reduced if m = l(w).

Theorem 2.3 Let w = s_{1} ⋯ s_{m} with s_{1}, …, s_{m} ∈ S. Then, there is a subset I = {i_{1} < i_{2} < ⋯ < i_{k}} of {1, …, m} with k = l(w) elements such that w = s_{i_{1}} ⋯ s_{i_{k}}.

This theorem is a direct consequence of the exchange lemma [Bki, Chap. IV, §1, Proposition 4] :

Lemma 2.4 Let w = s_{1} ⋯ s_{m} be a reduced decomposition (s_{1}, …, s_{m} ∈ S). Let s ∈ S. Then, one of the following assertion holds :

(i) l(sw) = l(w) + 1 and ss_{1} ⋯ s_{m} is a reduced decomposition of sw

(ii) l(sw) = l(w) − 1 and there exists j ∈ {1, …, m} such that s_{1} ⋯ s_{j−1}s_{j+1} ⋯ s_{m} is a reduced decomposition of sw and ss_{1} ⋯ s_{j−1}s_{j+1} ⋯ s_{m} is a reduced decomposition of w.

This lemma actually characterizes the Coxeter systems amongst the pairs (W, S), where S is a set of generators of order 2 of a group W [Bki, Chap. IV, §1, Théorème 1].

3 Real reflection groups

Let V be a finite dimensional real vector space. A reflection of V is an automorphism of order 2 whose set of fixed points is an hyperplane. A finite reflection group W in V is a finite subgroup of GL(V) generated by reflections.

The group W is crystallographic if there is a W -invariant Z-lattice of V, i.e., if there exists a free Z-submodule L of V stable under W such that the canonical map L ⊗_{Z} R → V is an isomorphism.

Note that this amounts to the existence of a W -stable Q-structure on V, i.e., a Q-subspace V_{Q} of V stable under W such that the canonical map V_{Q} ⊗_{Q} R → V is an isomorphism [Bki, Chap. VI, §2, Proposition 9].

Let A be the set of reflecting hyperplanes of W — i.e., the set of ker(s−1), where s is a reflection of W.

Then, V −⋃_{H∈A} H is in general non-connected : its connected components are the chambers of W.

Theorem 3.1 ([Bki, Chap. V, §3, Théorèmes 1 et 2]) The group W acts simply transitively on the set of chambers ; the closure of a chamber is a fundamental domain for the action of W on V.

Let C_{1} be a chamber and S the set of reflections with respect to the walls of C_{1} (a wall of C_{1} is an hyperplane in A whose intersection with the closure of C_{1} has codimension 1 in V).

Theorem 3.2 ([Bki, Chap. V, §3, Théorème 1]) The pair (W, S) is a Coxeter system.

Taking into account the choice of the chamber C_{1}, the chambers are now parametrized by W. The chamber C_{w} corresponding to w ∈ W is w(C_{1}).

A gallery of length n is a sequence D_{0}, …, D_{n} of adjacent chambers (i.e., the intersection of the closures of D_{i} and D_{i+1} has codimension 1 in V). The following result can be deduced from [Bki, Chap. V, §3, Théorème 1] :

Proposition 3.3 The minimal length of a gallery from C_{0} to C_{w} is l(w).

Example : the chamber system for the group A_{2}.

[figure: C_{1}; C_{s_{2}}; C_{s_{1}s_{2}}; C_{s_{1}s_{2}s_{1}}; C_{s_{2}s_{1}}; C_{s_{1}}; H_{2}; H_{1}]

Here, s_{i} is the orthogonal reflection with respect to H_{i}. The group W generated by s_{1} and s_{2} is a Coxeter group of type A_{2}.

4 Coxeter groups as reflection groups

Let S be a set and M = (m_{s,s′})_{s,s′∈S} a Coxeter matrix. Let V = R^{S} and denote by {e_{s}}_{s∈S} its canonical basis.

Define a bilinear form B_{M} on V by

B_{M}(e_{s}, e_{s′}) = − cos \frac{π}{m_{s,s′}} .

(Note that B_{M}(e_{s}, e_{s}) = 1).

Let ρ_{s} be the reflection in V given by

ρ_{s}(x) = x − 2B_{M}(e_{s}, x)e_{s}.

One has V = Re_{s} ⊕ H_{s}, where H_{s} is the hyperplane orthogonal to e_{s}.

Let W be the group with set of generators S and relations

s^{2} = 1

\underbrace{ss′ss′ ⋯}_{m_{ss′} terms} = \underbrace{s′ss′s ⋯}_{m_{ss′} terms} for those s, s′ ∈ S such that m_{s,s′} ≠ ∞.

Theorem 4.1 ([Bki, Chap. V, §4.3 et §4.4]) The map s ↦ ρ_{s} extends to an injective group morphism W → GL(V), the reflection representation of W. Furthermore, (W, S) is a Coxeter system.

When S is finite, B_{M} is positive definite if and only if W is finite.

Summarizing Theorems 3.2 and 4.1, we deduce

Theorem 4.2 The constructions of §3 and §4 give rise to inverse bijections between the set of conjugacy classes of finite subgroups of GL_{n}(R) generated by reflections and the set of those rank n Coxeter matrices giving rise to a finite Coxeter group.

5 Finite Coxeter groups

The classification of Coxeter graphs giving rise to irreducible finite Coxeter groups is the following [Bki, Chap. VI, §4, Théorème 1] (the number attached to the name of the diagram is the number of nodes of the diagram) :

\begin{matrix}A_{n} & [figure] \\ B_{n} & [figure] \\ D_{n} & [figure] \\ E_{6} & [figure] \\ E_{7} & [figure] \\ E_{8} & [figure] \\ F_{4} & [figure] \\ G_{2} = I_{2}(6) & [figure] \\ H_{3} & [figure: 5] \\ H_{4} & [figure: 5] \\ I_{2}(m) & [figure: m] (m = 5 or m ≥ 7) \end{matrix}

In the list above, all the groups are crystallographic except H_{3}, H_{4} and I_{2}(m), m = 5 or m ≥ 7.

6 Root systems and Weyl groups

Let V be a finite dimensional real vector space, Φ a finite subset of V and Φ^{∨} a finite subset of V^{∗} parametrized by Φ : Φ → Φ^{∨}, α ↦ α^{∨}.

Assume

(1) the vector space V is generated by Φ

(2) for all α ∈ Φ, we have ⟨α^{∨}, α⟩ = 2 and the reflection s_{α} : V → V, x ↦ x − ⟨α^{∨}, x⟩α stabilizes Φ

(3) we have α^{∨}(Φ) ⊂ Z for all α ∈ Φ

(4) for α ∈ Φ, we have 2α ∉ Φ.

Then, Φ is a root system in V (sometimes called reduced, because of (4)). Note that given Φ, there is at most one set Φ^{∨} parametrized by Φ with the required properties.

If Φ = Φ_{1} ∪ Φ_{2} and Φ_{i} (together with Φ_{i}^{∨} = {α^{∨}}_{α∈Φ_{i}}) is a root system in V_{i}, the subspace of V generated by Φ_{i}, for i ∈ {1, 2}, then we say that Φ is the direct sum of the root systems Φ_{1} and Φ_{2}. The root system Φ is irreducible if it is non-empty and it is not the direct sum of two non-empty root systems.

The Weyl group of the root system Φ is the subgroup of GL(V) generated by the reflections s_{α} for α ∈ Φ. Note that W is a crystallographic finite reflection group with Z-lattice the Z-submodule of V generated by Φ. A converse actually holds [Bki, Ch. VI, §2, Proposition 9] :

Proposition 6.1 Let W be a crystallographic reflection group in a finite dimensional real vector space V. Then, there is a root system Φ in V with Weyl group W.

Note that if W is irreducible, then the root system Φ is unique up to isomorphism if and only if W is not of type B_{n}, n ≥ 3 (cf Remark 2).

Let C be a chamber of W with walls L_{1}, …, L_{n}. Then, there is a unique root α_{i} ∈ Φ orthogonal to L_{i} and lying in the same half-space delimited by L_{i} as C.

The set Δ = {α_{i}}_{1≤i≤n} is called a basis of Φ.

Let Φ^{+} = {α ∈ Φ|α = ∑ n_{i}α_{i}, n_{i} ≥ 0} (the positive roots) and Φ^{−} = {α ∈ Φ|α = ∑ n_{i}α_{i}, n_{i} ≤ 0} (the negative roots).

Proposition 6.2 ([Bki, Chap. VI, §1, Théorèmes 2 et 3]) The set Δ is a basis of V and Φ = Φ^{+} ∪ Φ^{−}.

The Cartan matrix is (⟨α, β^{∨}⟩)_{α,β∈Δ}.

Define S = {s_{α}}_{α∈Δ}. Then, (W, S) is a Coxeter system.

The rank 2 root systems

[figure: Type A_{1} × A_{1}; Type A_{2}; α_{1}; α_{2}; −α_{1}; −α_{2}; α_{1}; α_{2}; α_{1} + α_{2}; −α_{1}; −α_{2}; −α_{1} − α_{2}]

[figure: Type B_{2} = C_{2}; Type G_{2}; α_{1}; α_{1} + α_{2}; 2α_{1} + α_{2}; α_{2}; −α_{2}; −α_{1}; −α_{1} − α_{2}; −2α_{1} − α_{2}; α_{1}; α_{2}; α_{1}+α_{2}; 2α_{1}+α_{2}; 3α_{1}+α_{2}; 3α_{1}+2α_{2}; −α_{1}; −α_{2}; −α_{1}−α_{2}; −2α_{1}−α_{2}; −3α_{1}−α_{2}; −3α_{1}−2α_{2}]

The action of W on Φ gives another interpretation of the length function [Bki, Chap. VI, §1, Corollaire 2 de la Proposition 17] :

Proposition 6.3 Let w ∈ W. The cardinality of Φ^{−} ∩ w(Φ^{+}) is the length l(w).

The set Φ^{∨} defines a root system in V^{∗} (the root system inverse or dual to Φ). There is an isomorphism of groups

W (Φ) → W (Φ^{∨}),
u ↦ ^{t}u^{−1}

sending s_{α} on s_{α^{∨}}. Through this isomorphism, W (Φ) operates on V^{∗}.

Remark 2 Note that the root systems (V, Φ) and (V^{∗}, Φ^{∗}) are not isomorphic in general : for example, the root system of type C_{n} is the inverse of the root system of type B_{n} ; when n ≥ 3, these root systems are not isomorphic.

7 Affine Weyl groups

Let Φ be a root system in a finite dimensional real vector space V. We construct a subgroup of the group Aff(V^{∗}) of affine transformations of V^{∗} as follows :

For α ∈ Φ and k ∈ Z, let H_{α,k} be the affine hyperplane of V^{∗} defined by

H_{α,k} = {x ∈ V^{∗}|⟨α, x⟩ = k}.

Let s_{α,k} be the orthogonal reflection with respect to H_{α,k} :

s_{α,k}(x) = x − (⟨α, x⟩ − k)α^{∨}.

The affine Weyl group associated to Φ is the subgroup W of Aff(V^{∗}) generated by the s_{α,k}, α ∈ Φ, k ∈ Z.

Let Q be the subgroup of Aff(V^{∗}) generated by the translations by elements of Φ^{∨}.

Proposition 7.1 ([Bki, Chap. VI, §2, Proposition 1]) We have W = Q ⋊ W.

An alcove is a connected component of V^{∗} − ⋃_{α∈Φ,k∈Z} H_{α,k} (note that the set {H_{α,k}}_{α∈Φ,k∈Z} is the set of reflecting hyperplanes of W).

Theorem 7.2 ([Bki, Chap. VI, §2.1]) The group W acts simply transitively on the set of alcoves. The closure of an alcove is a fundamental domain for the action of W on V^{∗}.

Let C be a chamber for W. Then, there is a unique alcove A ⊂ C such that 0 is in the closure of A.

Let S̃ be the set of reflections with respect to the walls of A. The pair (W, S̃) is Coxeter group.

If Φ is irreducible, there is a root α̃ = ∑ n_{i}α_{i} in Φ such that if β ∈ Φ, β = ∑ m_{i}α_{i}, then m_{i} ≤ n_{i} : α̃ is the highest root. This root is orthogonal to the wall of A which doesn’t contain 0 [Bki, Chap. VI, §1, Proposition 25].

The length of an element w ∈ W (relatively to S̃) is the minimal length of a gallery of alcoves from A to w(A).

Example : Type Ã_{2}.

[figure: H_{α_{2},0}; H_{α_{1},0}; H_{α̃,0}; A; 0; α_{1}; α_{2}; α̃]

The classification of the irreducible affine Weyl groups (or of their Coxeter graphs) is the following [Bki, Chap. VI, §4, Théorème 4] :

\begin{matrix}Ã_{1} & [figure: ∞] \\ Ã_{n} & [figure] (n ≥ 2) \\ B̃_{n} & [figure] (n ≥ 3) \\ C̃_{n} & [figure] (n ≥ 2) \\ D̃_{n} & [figure] (n ≥ 4) \\ Ẽ_{6} & [figure] \\ Ẽ_{7} & [figure] \end{matrix}

[figure: Ẽ_{8}; F̃_{4}; G̃_{2}]

8 Braid groups and Iwahori-Hecke algebras

Let (W, S) be a Coxeter system.

Definition 8.1 The braid monoid B_{W}^{+} associated with (W, S) is the monoid with set of generators {σ_{s}^{+}}_{s∈S} and relations :

\underbrace{σ_{s}^{+}σ_{s′}^{+}σ_{s}^{+}σ_{s′}^{+} ⋯}_{m_{ss′} terms} = \underbrace{σ_{s′}^{+}σ_{s}^{+}σ_{s′}^{+}σ_{s}^{+} ⋯}_{m_{ss′} terms} for s, s′ ∈ S such that m_{s,s′} is finite.

The braid group B_{W} associated with (W, S) is the group with set of generators {σ_{s}}_{s∈S} and relations :

\underbrace{σ_{s}σ_{s′}σ_{s}σ_{s′} ⋯}_{m_{ss′} terms} = \underbrace{σ_{s′}σ_{s}σ_{s′}σ_{s} ⋯}_{m_{ss′} terms} for s, s′ ∈ S such that m_{s,s′} is finite.

The following result is due to Deligne [De, Proposition 4.17] and Brieskorn-Saito [BrSa, Proposition 5.5] :

Theorem 8.2 Assume W is finite. Then, the morphism B_{W}^{+} → B_{W} given by σ_{s}^{+} ↦ σ_{s} is injective.

Thanks to this result, we can identify B_{W}^{+} with the submonoid of B_{W} consisting of those elements which can be written as products of generators σ_{s}, when W is finite.

Let p : B_{W}^{+} → W be given by σ_{s} ↦ s.

The exchange lemma (Lemma 2.4) has the following consequence [Bki, Chapitre IV, §1, Proposition 5] :

Proposition 8.3 Let w = s_{1} ⋯ s_{m} = s′_{1} ⋯ s′_{m} be two reduced expressions of w ∈ W. Then, σ_{s_{1}}^{+} ⋯ σ_{s_{m}}^{+} = σ_{s′_{1}}^{+} ⋯ σ_{s′_{m}}^{+}.

This allows the construction of a very nice section q of p (q is not multiplicative !) : given w ∈ W and w = s_{1} ⋯ s_{m} a reduced expression of w, we put q(w) = σ_{s_{1}}^{+} ⋯ σ_{s_{m}}^{+}. Thanks to the last proposition, this is independent of the choice of the reduced expression of w. We have pq(w) = w.

Let S̄ be the set of equivalence classes for the relation defined by : s, t ∈ S are equivalent if there is a sequence t = t_{0}, t_{1}, …, t_{m} = s such that m(t_{i}, t_{i+1}) is finite and odd. For s ∈ S, we denote by s̄ its class in S̄.

One can then read off conjugacy amongst generators of W and B_{W} from the Coxeter diagram. Two elements s, t ∈ S are conjugate in W if and only if s̄ = t̄. Similarly, σ_{s} and σ_{t} are conjugate in B_{W} if and only if s̄ = t̄. So, we have :

Proposition 8.4 The map σ_{s} ↦ 1s̄ extends to a group morphism B_{W} → ZS̄. Its kernel is the derived subgroup of B_{W}, i.e., this morphism identifies the largest abelian quotient B_{W}^{ab} of B_{W} with ZS̄. Similarly, W^{ab} is isomorphic to (Z/2Z)S̄.

Examples (using the notations from the examples of §2).

• 𝔖_{n} : S̄ = {s̄_{1}}.

• B_{n} : S̄ = {s̄_{0}, s̄_{n−1}} (n ≥ 2).

• I_{2}(m) : S̄ = {t̄_{1}} for odd m, S̄ = {t̄_{1}, t̄_{2}} for even m.

Let {qs̄}_{s̄∈S̄} and {q′ ; s̄}_{s̄∈S̄} be two sets of indeterminates. Let Õ = Z[qs̄, q′ ; s̄]_{s̄∈S̄}. The Iwahori-Hecke algebra H̃ = H̃(W) of (W, S) is the Õ-algebra with

generators T_{s}, s ∈ S, and relations

(T_{s} − qs̄)(T_{s} − q′ ; s̄) = 0 for s ∈ S

\underbrace{T_{s}T_{s′}T_{s}T_{s′} ⋯}_{m_{ss′} terms} = \underbrace{T_{s′}T_{s}T_{s′}T_{s} ⋯}_{m_{ss′} terms} for s, s′ ∈ S such that m_{s,s′} is finite.

This is the quotient of the monoid algebra ÕB_{W}^{+} by the ideal generated by the elements (T_{s} − qs̄)(T_{s} − q′ ; s̄) for s ∈ S. Let T_{w} be the image of σ_{w}^{+} for w ∈ W.

The next theorem shows that H̃ is a deformation of ZW [Bki, Chap. IV, §2, Exercice 23] :

Theorem 8.5 The algebra H̃ is free over Õ, with basis {T_{w}}_{w∈W}. The morphism H̃ ⊗Õ Õ/(qs̄ − 1, q′ ; s̄ + 1)_{s̄∈S̄} → ZW, T_{w} ⊗ 1 ↦ w, is an isomorphism.

We assume from now on that W is finite.

Let S′ be a subset of S and W′ be the subgroup of W generated by S′. Then, by [De],

• the submonoid of B_{W}^{+} generated by {σ_{s}^{+}}_{s∈S′} is isomorphic to B_{W′}^{+}, • the subgroup of B_{W} generated by {σ_{s}}_{s∈S′} is isomorphic to B_{W′},

• the specialization of the subalgebra of H̃(W) generated by {T_{s}}_{s∈S′} obtained by sending to 0 those parameters not associated to elements s̄, s ∈ S′, is isomorphic to the specialization of H̃(W′) given by identifying those parameters associated to elements of S̄′ which become equal in S̄.

In several applications, the Iwahori-Hecke algebra arises with invertible parameters. Then, without loss of generality, one may assume one of the two

parameters qs̄, q′ ; s̄ to be −1. So, let O = Õ[q−1 ; s̄]_{s̄∈S̄}/(q′ ; s̄ + 1)_{s̄∈S̄} ≃ Z[qs̄, q−1 ; s̄]_{s̄∈S̄}

and H = H̃ ⊗Õ O.

The Iwahori-Hecke algebra has a trace map τ : H → O given by τ(T_{w}) = δ_{1,w} (i.e., we have τ(hh′) = τ(h′h) for h, h′ ∈ H).

Denote by “ind” the one-dimensional representation H → O given by ind(T_{s}) = qs̄.

Proposition 8.6 Given w, w′ ∈ W, one has τ(T_{w}T_{w′}) = δ_{w^{−1},w′}ind(T_{w}).

This means that the set {ind(T_{w})^{−1}T_{w^{−1}}}_{w∈W} is the dual basis of {T_{w}}_{w∈W} with respect to τ.

More conceptually, the trace τ gives a structure of symmetric algebra to H, i.e., the morphism :

H → Hom_{O}(H, O)
h ↦ (h′ ↦ τ(hh′))

is an isomorphism.

Together with the fact that H is a deformation of ZW, this explains the structure of H over an algebraic closure K of the field of fractions of O (Tits’ deformation theorem) [Bki, Chap. IV, §2, Exercice 27] :

Theorem 8.7 The algebra H ⊗_{O} K is semi-simple and isomorphic to KW.

Much more precise is the following rationality theorem (Benard, Springer, Benson-Curtis, Hoefsmit, Lusztig..., cf [Ge]) :

Theorem 8.8 Assume W is a finite Weyl group. Then, the algebra QW is isomorphic to a direct product of matrix algebras over Q and the algebra H ⊗_{O} Q(\sqrt{qs̄})_{s̄∈S̄} is isomorphic to a direct product of matrix algebras over Q(\sqrt{qs̄})_{s̄∈S̄}.

The theorem above generalizes to finite Coxeter groups : if W is a finite reflection group over K ⊂ R, then KW is isomorphic to a product of matrix algebras over K and H ⊗_{O} K(\sqrt{qs̄})_{s̄∈S̄} is isomorphic to a direct product of matrix algebras over K(\sqrt{qs̄})_{s̄∈S̄}.

9 Pseudo-reflection groups

Let V be a finite dimensional vector space over a characteristic zero field K. A pseudo-reflection of V is an automorphism of finite order whose set of fixed points is an hyperplane.

Let G be a finite subgroup of GL(V).

We denote by S(V) the symmetric algebra of V.

The following theorem is due to Shephard-Todd and Chevalley. It shows that the regularity of the ring of invariants S(V)^{G} characterizes pseudoreflection groups [Bens, Theorem 7.2.1] :

Theorem 9.1 The following assertions are equivalent :

(i) The algebra S(V)^{G} is a polynomial algebra.

(ii) The group G is generated by pseudo-reflections.

(iii) The S(V)^{G}[G]-module S(V) is free of rank one.

When K ⊂ R, a pseudo-reflection is actually a reflection. When K = C, a pseudo-reflection need not have order 2 ; a group generated by complex pseudo-reflections is then called a complex reflection group.

The rationality theorem 8.8 for representations of Weyl groups or finite Coxeter groups extends to pseudo-reflection groups : the group algebra of G over K is a direct product of matrix algebras over K [Bena, Bes].

The irreducible complex reflection groups have been classified by Shephard and Todd [ShTo]. There are two infinite series : the groups A_{n} ≃ 𝔖_{n+1}, the groups G(p, q, n) and 34 exceptional groups (the dimension of an exceptional group is at most 8).

Let us describe the groups G(p, q, n) (p > 1, q ≥ 1, n ≥ 1 and q|p). It turns out that these groups have nice presentations, generalizing in some sense the presentation of Coxeter groups and sharing some of their properties. In particular, these groups have a presentation given by a set S consisting of n or n + 1 pseudo-reflections and two kinds of relations :

• braid relations (homogeneous relations)

• finite order relations.

The group given by the same presentation, but without the finite order relations can be seen as an analog of the braid group defined in §8 for real reflection groups. We will come back to this in §10.

9.1 G(p, 1, n)

First, G(p, 1, n) is the group of n by n monomial complex matrices whose non-zero entries are p-th roots of unity. This group has a semi-direct product decomposition G(p, 1, n) = (Z/pZ)^{n} ⋊ 𝔖_{n} ≃ (Z/pZ) ≀ 𝔖_{n}, where 𝔖_{n} is the subgroup of permutation matrices and (Z/pZ)^{n} is the subgroup of diagonal matrices.

Let s_{0} = diag(ζ, 1, …, 1), where ζ is a primitive p-th root of unity. Keeping the notations of §2, Example (i), one sees that G(p, 1, n) is generated by the set of pseudo-reflections {s_{0}, s_{1}, …, s_{n−1}}. They satisfy the following relations :

braid relations \begin{cases}s_{0}s_{1}s_{0}s_{1} = s_{1}s_{0}s_{1}s_{0} \\ s_{i}s_{j} = s_{j}s_{i} & if |i − j| > 1 \\ s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} & for i ≥ 1 \end{cases}

finite order relations \begin{cases}s_{0}^{p} = 1 \\ s_{i}^{2} = 1 for i ≥ 1. \end{cases}

Actually, this gives a presentation for G(p, 1, n) by generators and relations. A convenient way to encode the relations is to use a generalization of the Coxeter diagrams :

[figure: p; s_{0}; s_{1}; s_{2}; s_{3}; s_{n−2}; s_{n−1}]

Note that G(2, 1, n) = B_{n} and the presentation above is the Coxeter presentation.

Now, for q|p, we define G(p, q, n) as the subgroup of G(p, 1, n) consisting of matrices where the product of the non-zero entries is a (p/q)-th root of unity.

9.2 G(p, p, n)

Let us now look at G(p, p, n). It is generated by the set of pseudo-reflections {s′_{1}, s_{1}, …, s_{n−1}} where s′_{1} = s_{0}s_{1}s_{0}^{−1}. They satisfy the following relations :

braid relations \begin{cases}s_{i}s_{j} = s_{j}s_{i} & if |i − j| > 1 \\ s′_{1}s_{i} = s_{i}s′_{1} & for i ≥ 3 \\ s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} & for i ≥ 1 \\ s′_{1}s_{2}s′_{1} = s_{2}s′_{1}s_{2} \\ s_{2}s′_{1}s_{1}s_{2}s′_{1}s_{1} = s′_{1}s_{1}s_{2}s′_{1}s_{1}s_{2} \\ \underbrace{s_{1}s′_{1}s_{1}s′_{1} ⋯}_{p terms} = \underbrace{s′_{1}s_{1}s′_{1}s_{1} ⋯}_{p terms} \end{cases}

finite order relations \begin{cases}s′_{1}^{2} = 1 \\ s_{i}^{2} = 1 for i ≥ 1. \end{cases}

This gives a presentation of G(p, p, n) by generators and relations. The relations may be encoded in the following diagram :

[figure: s′_{1}; s_{1}; s_{2}; s_{3}; s_{n−2}; s_{n−1}; p]

Note that G(p, p, 2) = I_{2}(p) and the presentation above is a Coxeter presentation. Also, G(2, 2, n) = D_{n} and the presentation above is a Coxeter presentation.

9.3 G(p, q, n)

Finally, let us consider G(p, q, n) for q|p, q ≠ p and q ≠ 1. We put d = p/q. This group is generated by the set of pseudo-reflections {s′_{0}, s′_{1}, s_{1}, …, s_{n−1}} where s′_{0} = s_{0}^{q}. They satisfy the following relations :

braid relations \begin{cases}s_{i}s_{j} = s_{j}s_{i} & if |i − j| > 1 \\ s′_{0}s_{i} = s_{i}s′_{0} & for i ≥ 2 \\ s′_{1}s_{i} = s_{i}s′_{1} & for i ≥ 3 \\ s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} & for i ≥ 1 \\ s′_{1}s_{2}s′_{1} = s_{2}s′_{1}s_{2} \\ s′_{0}s′_{1}s_{1} = s′_{1}s_{1}s′_{0} \\ s_{2}s′_{1}s_{1}s_{2}s′_{1}s_{1} = s′_{1}s_{1}s_{2}s′_{1}s_{1}s_{2} \\ \underbrace{s_{1}s′_{0}s′_{1}s_{1}s′_{1}s_{1} ⋯}_{q+1 terms} = \underbrace{s′_{0}s′_{1}s_{1}s′_{1}s_{1}s′_{1} ⋯}_{q+1 terms} \end{cases}

finite order relations \begin{cases}s′_{0}^{d} = 1 \\ s′_{1}^{2} = 1 \\ s_{i}^{2} = 1 for i ≥ 1. \end{cases}

We have obtained a presentation of G(p, q, n) by generators and relations which we encode in the following diagram :

[figure: d; s′_{0}; s′_{1}; s_{1}; s_{2}; s_{3}; s_{n−2}; s_{n−1}; q+1]

10 Topological construction of braid groups and Iwahori-Hecke algebras

Let V be a finite dimensional complex vector space and G a finite subgroup of GL(V) generated by pseudo-reflections. Let A be the set of reflecting hyperplanes of G and X = V −⋃_{H∈A} H. Let ρ : X → X/G be the projection map.

The following result is due to Steinberg [St] :

Theorem 10.1 The group G acts freely on X, i.e., ρ is an unramified Galois covering.

Let x_{0} ∈ X. The braid group associated to G is B_{G} = Π_{1}(X/G, ρ(x_{0})) and the pure braid group associated to G is P_{G} = Π_{1}(X, x_{0}). Then, thanks to Steinberg’s theorem, we have an exact sequence :

0 → P_{G} ⟶^{ρ_{∗}} B_{G} → G → 1.

10.1 The real case

Assume G is the complexification of a real reflection group, i.e., there is a real vector space V′ with V = V′ ⊗_{C} R and such that G is a subgroup of GL(V′). Let C_{1} be a chamber of G (a connected component of V′ −⋃_{H∈A} H ∩V′) and take x_{0} ∈ C_{1}. Let S be the set of reflections of G with respect to the walls of C_{1}. For s ∈ S, let γ_{s} be the path [0, 1] → X defined by

γ_{s}(t) = \frac{x_{0} + s(x_{0})}{2} + \frac{x_{0} − s(x_{0})}{2}e^{iπt}.

Let τ_{s} be the class in B_{G} of ρ(γ_{s}).

Brieskorn [Br] and Deligne [De] have proven the following theorem :

Theorem 10.2 The map σ_{s} ↦ τ_{s} induces an isomorphism B_{G} → B_{G}.

10.2 The complex case

Let H ∈ A. Let e_{H} be the order of the pointwise stabilizer of H in G. This is a cyclic group, generated by a pseudo-reflection s with non-trivial eigenvalue exp(2iπ/e_{H}). Let x_{H} ∈ X. Let y_{H} be the intersection of H with the affine line containing x_{H} and s(x_{H}). We assume x_{H} is close enough to H so that the closed ball with center y_{H} and radius ||x_{H} − y_{H}|| does not intersect any H′, H′ ∈ A, H′ ≠ H. Let α be a path from x_{0} to x_{H} in X. Let λ be the path in X from x_{H} to s(x_{H}) defined by

λ(t) = y_{H} + (x_{H} − y_{H})e^{2iπt/e_{H}}.

We define the path γ from x_{0} to s(x_{0}) by

γ(t) = \begin{cases}α(3t) & for 0 ≤ t ≤ 1/3 \\ λ(3t − 1) & for 1/3 < t ≤ 2/3 \\ s(α(3 − 3t)) & for 2/3 < t ≤ 1 \end{cases}

Let τ be the class of γ in B_{G} : τ is called a generator of the monodromy associated to s (or to H). The image of τ in G is s.

Theorem 10.2 has a counterpart for complex reflection groups, based on a case by case study [BrMaRo] (six of the irreducible exceptional complex reflection groups are not covered by this approach).

Let us explain this for the group G = G(p, q, n).

Theorem 10.3 Assume G = G(p, q, n). Then, for every s ∈ S, there is a generator of the monodromy τ_{s} associated to s, such that the group B_{G} has a presentation with set of generators {τ_{s}}_{s∈S} and relations the braid relations.

10.3 Iwahori-Hecke algebras

An analogue of Proposition 8.4 for complex reflection groups is : B_{G}^{ab} is isomorphic to Z^{A/G}.

For C ∈ A/G and H ∈ C, we put e_{C} = e_{H}. Let Õ be the polynomial ring over the integers on the set of variables {q_{C,j}}_{C∈A/G,0≤j≤e_{C}−1}. We put q_{H,j} = q_{C,j} for H ∈ C and C ∈ A/G.

For H ∈ A, let τ_{H} be a generator of the monodromy associated to H. Let H̃ be the quotient of the group algebra ÕB_{G} by the ideal generated by the (τ_{H} − q_{H,0})(τ_{H} − q_{H,1}) ⋯ (τ_{H} − q_{H,e_{H}−1}), for H ∈ A. Then, H̃ is the Iwahori-Hecke algebra associated to G.

When G is the complexification of a real reflection group, then Theorem 10.2 induces an isomorphism with the Iwahori-Hecke algebra associated with the Coxeter system (G, S).

When G = G(p, q, n), the algebra H̃ is a deformation of the group algebra ÕG (it is free over Õ, with rank |G|) [Ar].

References

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