On the linearity of Artin braid groups

François DIGNE

0. Introduction and notation. The aim of this paper is to prove that all Artin groups of crystallographic type have a faithful representation of dimension the number of reflections of the associated Coxeter group. We also give some properties of the representation we construct. The starting point of this paper is the proof given by Krammer [K] of the linearity of the classical braid groups. But our method goes along different lines as we first construct a priori an action of an Artin monoid on the closed subsets of positive roots and then construct the desired representation. The faithfulness criterion which we use is that of Krammer. Note that, as any Artin group of spherical type can be embedded in an Artin group of type A_{n}, D_{n} or E_{n}, as far as only linearity is concerned it is sufficient to deal with these cases. Here for all crystallographic types we prove more: we give a faithful linear representation in a vector space of dimension equal to the number of reflections of the Coxeter group.

François Digne

The main lines proceed as follows: in section 1 we recall some facts about closed sets of roots in a root system and about elementary properties of braid monoids. In section 2 we define an action of the braid monoid associated to a crystallographic Coxeter group on the set of closed subsets of positive roots (in fact this action is the restriction of an action on all subsets of positive roots but we do not need this fact here). This action satisfies a crucial property (2.2 (iii)) which will be the key to prove the faithfulness of the representations in section 3. Following Krammer’s ideas, we want to define a representation of the braid monoid B^{+} on the ring of polynomials in one variable ℝ[t] where the generators of B^{+} act by matrices with positive coefficients modulo t. This is done in section 3, first for groups with a simply laced diagram. This representation is defined in such a way that its specialisation at t = 0 is strongly linked with the action of B^{+} on closed subsets of positive roots, which, using the above mentionned crucial property, will imply the faithfulness of the representation by Krammer’s criterion. We also prove in section 3 a unicity property for the representation, once the specialisation for t = 0 is known, under some additonal property (theorem 3.8). The representation of B^{+} thus obtained extends to B. We then deal with the groups with a non simply laced diagrams using the fact that such a group can be realized as the group of fixed points of a graph automorphism in a group of simply laced type: as our unicity property allows us to extend the representation to the semidirect product of an Artin group by a graph automorphism, we can make the group of fixed points act on the subspace of fixed vectors by the automorphism. We show that this gives still a faithful representation using the results of section 2. In section 4 we give some formulas and in section 5 we give some properties of our representations.

In the course of writing this paper we have learned that results similar to our section 3 for the case of Artin groups of type A_{n}, D_{n} or E_{n} have been proved simultaneously by Cohen and Wales ([CW]).

Our notation will be the following. We denote by (W, S) a crystallographic Coxeter system, by B^{+} the associated Artin monoid and by B the corresponding Artin group. The canonical generating sets of either B or B^{+} will be identified and denoted by S. This set is in one-to-one correspondence with S. We denote by p the natural group homomorphism from B to W. It has a set theoretic section obtained by mapping an element of W to the lifting in B^{+} of any of its reduced expressions. The image of this section will be denoted by B_{red}.

As our main results for an arbitrary W can be deduced from the case when W is irreducible, we shall always assume that we are in the latter case.

1. Closed subsets of reflections; Artin monoids. First we recall some properties of root systems. The references for these results are [Bbki], [Dy] and [Pa]. Let Φ be a finite reduced root system with a given basis of simple roots, defining a set Φ^{+} of positive roots, such that W is the Weyl group of this root system: the elements of S are the reflections defined by the simple roots. We denote by R the set of reflections of W; this set is in oneto-one correspondence with Φ^{+}. The image of a root α ∈ Φ under the action of w ∈ W will be denoted by w(α). If r is the reflection corresponding to α, the root w(α) corresponds to the conjugate wrw^{−1} which we denote by ^{w}r.

For any w ∈ W we put N_{Φ}(w) = Φ^{+} ∩ w(−Φ^{+}) and N(w) = {r ∈ R | l(rw) < l(w)}. It is known that the map which associates a reflection to a root induces a one-to-one correspondence between these two sets. We denote by l(w) the length of w ∈ W with respect to the generating set S, i.e., the minimal number of factors in a decomposition of w into a product of elements of S. It is known (cf. [Bbki, IV, 1.4]) that l(w) = |N(w)| = |N_{Φ}(w)|. Before giving more properties of these sets we recall the following definition.

Definition 1.1. A set A ⊂ Φ is said to be closed if for any roots α and β in A such that α + β is a root then α + β is in A.

linearity of Artin groups

More generally it can be proved that a closed set which contains α and β contains any positive combination of them which is a root. We shall also say that a subset of R is closed when the corresponding subset of positive roots is closed.

The following proposition gives the main properties of the sets N(w) and N_{Φ}(w) (cf.

[Dy] and [Pa]).

Proposition 1.2.

(i) For any w ∈ W the set N(w) is closed and its complement in R is closed. (ii) Any closed subset of R which has a closed complement in R is equal to N(w) for a

unique w ∈ W. (iii) For w and w′ in W we have N(ww′) = (N(w)) ∔ (^{w}N(w′)), where ∔ is the symmetric

difference operator. (iv) For w and w′ in W, we have N(w′) ⊂ N(w) if and only if there exists w′′ ∈ W such

that w = w′w′′ and l(w) = l(w′) + l(w′′).

We shall use (iv) in the above lemma in the equivalent form: if b and b′ are in B_{red} such that N(p(b′)) ⊂ N(p(b)) then b′ is a left divisor of b in B^{+}.

We now recall some known facts on Artin monoids. The references for these results are [D] and [Mi].

Proposition 1.3.

(i) For any b ∈ B^{+} there exists a unique element α(b) ∈ B_{red} maximal for divisibility among the left divisors of b.

(ii) For any b and b′ in B^{+} we have α(bb′) = α(bα(b′)). (iii) Any element in B^{+} can be written uniquely as a product x_{1} … x_{n} of elements of B_{red}

where x_{i} = α(x_{i} … x_{n}). (iv) B^{+} embeds in B and any element of B can be written a^{−1}b with a and b in B^{+}.

A decomposition as in (iii) above will be called a normal form.

We shall also use the following lemma ([D 1.14] or [Mi 1.4, 2.5 and proof of 2.6]) which characterizes the sets of divisors of an element of B^{+}.

Lemma 1.4. Let A be a finite non-empty subset of B^{+}. Then A is the set of all left divisors of some element of B^{+} if and only if it has the two following properties:

(i) A is stable by left divisibility. (ii) If x ∈ A, s, t ∈ S are such that xs ∈ A and xt ∈ A then xΔ_{st} ∈ A where Δ_{st} is the

longest element of B_{red} involving only s and t.

As B_{red} is the set of all left divisors of its longest element Δ, we get by 1.4:

Corollary 1.5. If x ∈ B_{red} and s, t ∈ S are such that xs and xt are in B_{red} then xΔ_{st} is in B_{red}.

2. An action of B^{+} on closed sets of reflections. We now will define an action of B^{+} on the set of closed subsets of R (or of Φ^{+}). The first result we need is the following

Lemma 2.1. Let A be a closed subset of Φ^{+}, then there exists a (unique) b ∈ B_{red} such that an element w ∈ W satisfies N_{Φ}(w) ⊂ A if and only if w = p(b′) for some left divisor b′ of b. In particular N_{Φ}(p(b)) is the greatest (for inclusion) subset of A of the form N_{Φ}(w) with w ∈ W.

Proof: Let E = {b ∈ B_{red} | N_{Φ}(p(b)) ⊂ A}; then E is finite, non-empty and stable by left divisibility. Assume that for some x ∈ E and some s, t ∈ S we have xs ∈ E and xt ∈ E. Then we have N_{Φ}(p(xs)) = N_{Φ}(p(x)) ∪ p(x)(N_{Φ}(p(s))) as xs ∈ B_{red} and similarly for xt. But N_{Φ}(p(s)) is equal to the singleton {α} where α is the simple root corresponding to s and N_{Φ}(p(t)) is reduced to the simple root β corresponding to t. As A is closed it has to contain all the positive roots corresponding to the dihedral subgroup generated by ^{p(x)}(p(s)) and p(x)(p(t)) as they are positive combinations of p(x)(α) and p(x)(β). This set of roots is exactly p(x)(N_{Φ}(p(Δ_{st}))). So A contains N_{Φ}(p(x)) ∪ p(x)(N_{Φ}(p(Δ_{st}))) = N_{Φ}(p(xΔ_{st})), by corollary 1.5 and 1.2 (iii), so that xΔ_{st} is in E. By 1.4 we see that there exists b ∈ B_{red} such that E is the set of left divisors of b.

We are now ready to define the wanted action.

Proposition 2.2.

(i) Let A ⊂ R be a closed set and let w ∈ W; then there exists a (unique) greatest closed set E ⊂ R such that N(w) ⊂ E ⊂ ^{w}A ∪ N(w).

(ii) The map which to b ∈ B_{red} and to a closed subset A of R associates the greatest closed subset of ^{w}A ∪ N(w) containing N(w), where w = p(b), can be uniquely extended in an action of the monoid B^{+} on the set of closed subsets of R. This action denoted by A ↦ b.A satisfies

(iii) For any x ∈ B^{+} and any closed subset A ⊂ R, if the greatest element of {N(w) | w ∈ W } which is a subset of A is N(p(y)) then the greatest such subset in x.A is N(p(α(xy))).

This action has been first obtained by Krammer when W is of type A_{n} as a consequence of his construction of a faithful representation of the braid group. Proof: We first prove the following elementary lemma on root systems.

Lemma 2.3. Let α, β, and γ be positive roots and n a positive integer such that β + γ and β + γ + nα are roots; then there exists an integer m with 0 ≤ m ≤ n such that β + mα and γ + (n − m)α are roots.

Proof: We first prove the result when n = 1. Note that if β or γ is equal to α there is nothing to prove. We now assume that β ≠ α and γ ≠ α. We denote by ( , ) a W -invariant positive inner product on the root system. Recall (cf. [Bbki, VI, 1.3, corollary of theorem 1]) that if the inner product of two non-collinear roots is strictly positive (resp. negative) then their difference (resp. their sum) is a root. So we can assume that (α, β) and (α, γ) are non-negative. But then (β + γ, α + β + γ) > 0, so one of (β, α + β + γ) or (γ, α + β + γ) has to be strictly positive so either α + β + γ − β or α + β + γ − γ is a root.

We now prove the lemma by induction on n. If β+γ+nα is a root then β+γ+(n−1)α is a root by [Bbki, VI, 1.3, proposition 9], so by induction there exists 0 ≤ m ≤ n − 1 such that β′ = β + mα and γ′ = γ + (n − 1 − m)α are roots. As β′ + γ′ + α is a root we can use the result in the case n = 1 to complete the proof.

The following lemma will give the inductive step to prove 2.2 (i)

Lemma 2.4. If A ⊂ Φ^{+} is closed, then for any simple root α the set {α} ∪ {β ∈ A | ∀n ∈ ℕ, β + nα ∈ Φ^{+} ⇒ β + nα ∈ A} is the greatest closed subset of {α} ∪ A containing α (in particular such a greatest subset exists).

Proof of the lemma: Let E_{0} = {β ∈ A | ∀n ∈ ℕ, β + nα ∈ Φ^{+} ⇒ β + nα ∈ A}. Any subset E of A such that E ∪ {α} is closed has to be a subset of E_{0}, so proving the lemma is equivalent to proving that E_{0} ∪ {α} is closed. Consider two roots β and γ in E_{0} ∪ {α} such that β + γ ∈ Φ^{+}. If β ∈ E_{0} and γ = α then by definition we have β + α ∈ E_{0}. If β and γ are both in E_{0} we want to show that β + γ is in E_{0}. If β + γ + nα ∈ Φ^{+} for some positive integer n then, by lemma 2.3, there exists an integer m with 0 ≤ m ≤ n such that β +mα ∈ Φ^{+} and γ +(n−m)α ∈ Φ^{+}; but then we have β +mα ∈ A and γ +(n−m)α ∈ A by the defining property of E_{0}, and as their sum is a root it has to be in A as A is closed, so β + γ + nα is in A, which shows that β + γ satisfies the defining property of E_{0}.

We now prove the proposition. We prove (i) by induction on the length of w. If w = 1 there is nothing to prove. Assume that w = sw′ with s ∈ S and l(w′) = l(w) − 1, and assume that the statement is true for w′. Then we have a greatest closed set E′ ⊂ w′A∪N(w′) containing N(w′). If E is any closed subset of ^{w}A∪N(w) containing N(w) then sN(w) ⊂ ^{s}E ⊂ ^{w′}A∪^{s}N(w), so that ^{s}E\{s} is a closed set containing N(w′) = ^{s}N(w)\{s} and contained in ^{w′}A ∪ N(w′), whence ^{s}E\{s} ⊂ E′, by definition of E′, and in turn E ⊂ ^{s}E′ ∪ {s}. As ^{s}E′ ∪ {s} ⊂ ^{w}A ∪ ^{s}N(w′) ∪ {s} = ^{w}A ∪ N(w), we have proved that a set E ⊂ R satisfies (i) if and only if it is the greatest closed subset of ^{s}E′ ∪ {s} containing N(w). As by conjugation by s and taking the union with {s}, we have N(w) ⊂ ^{s}E′ ∪ {s} and N(w) is closed and contains s, we get (2.5) A subset E ⊂ R satisfies 2.2 (i) if and only if it is the greatest closed subset of sE′ ∪ {s} containing s.

As s ∉ N(w′) we have ^{s}N(w′) ⊂ ^{s}E′\{s} ⊂ ^{w}A ∪ N(w) (note that if s ∉ E′ we have put ^{s}E′\{s} = ^{s}E′). As ^{s}E′\{s} is closed we can apply 2.4 to the set of positive roots corresponding to ^{s}E′\{s}. We get a greatest closed subset E of ^{s}E′ ∪ {s} containing s, so we have proved (i).

Now property 2.5 shows that, if to b ∈ B_{red} and A a closed subset of R, we associate the greatest closed subset as in (i) relative to w = p(b), denoted by b.A, we have b.(b′.A) = (bb′).A if bb′ ∈ B_{red}. This proves (ii) as we have a presentation of B^{+}, taking B_{red} as a generating set and the (partially defined) product in B_{red} as relations, so that any map from B_{red} to a monoid compatible with the product on B_{red} extends to B^{+}.

We now prove (iii). The proof we give here is very similar to that in [K, 5.6]. Using property 1.3 (ii) it is sufficient to prove (iii) for x ∈ B_{red}. As N(p(x)) ⊂ x.A the maximal element b ∈ B_{red} such that N(p(b)) ⊂ x.A has to be a left multiple of x (cf. 1.2 (iv)). We write this element b = xb′. So we have N(p(xb′)) = N(p(x)) ∪ ^{p(x)}N(p(b′)) ⊂ x.A ⊂ p(x)A ∪ N(p(x)), which implies N(p(b′)) ⊂ A as N(p(x)) is disjoint from ^{p(x)}N(p(b′)) because xb′ ∈ B_{red}. So by definition of y, we get that b′ divides y in B^{+} so that b = xb′ divides xy, so divides α(xy). On the other hand α(xy) can be written xy′ with y′ dividing y, so N(p(y′)) ⊂ N(p(y)) ⊂ A and N(p(α(xy))) = N(p(x)) ∪^{p(x)}N(p(y′)) ⊂ ^{p(x)}A ∪ N(p(x)), so that by definition of x.A we have N(p(α(xy))) ⊂ x.A. This proves the result.

In the rest of this paper we shall use a different notation for elements of S and of S. We use bold letters for elements of S, the image in W of s ∈ S being denoted by s. This will allow us to avoid using p in the notation.

Corollary 2.6. For any closed subset A ⊂ R and any s ∈ S corresponding to a simple root α, the subset s.A corresponds to the set of positive roots {α} ∪ {β ∈ Φ^{+} | ∀n ∈ ℕ, β + nα ∈ Φ^{+} ⇒ β + nα ∈ s(A_{Φ})}, where A_{Φ} is the set of positive roots corresponding to A.

Proof: This is the first case of the induction in the proof of 2.2 (i): we are in the case where w′ = 1 so that E′ = A and we have to apply lemma 2.4 to the set of positive roots s(A_{Φ}\{α}). We get the result if we note that if β ∈ Φ^{+} satisfies the condition as stated in the lemma then it has to be in s(A_{Φ}\{α}) by the case n = 0 of this condition.

3. Faithful representations of B and B^{+}. We now want to define a faithful representation of B. The criterion for faithfulness which we will use is the one introduced by Krammer. Note first that a representation of B is faithful if and only if its restriction to the monoid B^{+} is faithful, which is a consequence of 1.3 (iv). Let us recall the criterion (cf. [K, 2.1]).

Faithfulness criterion 3.1. Let us consider an action of B^{+} on some set E and denote by ρ(b) the permutation of E made by b ∈ B^{+}; assume that E contains a family of nonempty disjoint subsets C_{y} indexed by the elements y ∈ B_{red}, and such that ρ(x)(C_{y}) ⊂ C_{α(xy)} for any x ∈ B^{+}, then ρ is injective.

Property 1.3 (ii) implies by an induction argument that it is sufficient to check this criterion for x ∈ S. Following Krammer we shall define a representation of B on a vector space over the field of rational functions in one indeterminate ℝ(t). By 1.3 (iv) it is equivalent to define a representation of B or a representation of B^{+} in which all elements act by invertible linear operators.

Let V be a vector space over this field with a basis in one-to-one correspondence with R, which we denote by (e_{r})_{r∈R}. For the sake of simplicity we shall sometimes abuse notation and write e_{β} instead of e_{r} if β is the positive root corresponding to the reflection r. For any subset A ⊂ R we put D_{A} = ∑_{r∉A} ℝ_{>0}e_{r} + ∑_{r∈R} tℝ[t]e_{r} and for y ∈ B_{red} we denote by C_{y} the union of all sets D_{A} such that the greatest subset of the form N(w) contained in A is N(p(y)). We want a representation which satisfies criterion 3.1 for these sets C_{y}, which are clearly non-empty and disjoint. Using property 2.2 (iii) and the remark below 3.1 the property in the faithfulness criterion is a consequence of

For any s ∈ S and any closed A ⊂ R we have ρ(s)(D_{A}) ⊂ D_{s.A}.  (3.2)

We shall first define a representation of B^{+} in the free ℝ[t] submodule of V with basis (e_{r})_{r∈R}, satisfying (3.2).

As we work over the ring ℝ[t] we can put t = 0 in the coefficients of the representation. We then obtain a representation ρ_{0} of B^{+}. As D_{A} can be defined as the set of vectors which modulo t are positive linear combinations of the e_{r} with r ∉ A, we see that property (3.2) is equivalent to

For any s ∈ S and any closed A ⊂ R we have ρ_{0}(s)(∑_{r∉A} ℝ_{>0}e_{r}) ⊂ ∑_{r∉s.A} ℝ_{>0}e_{r}. (3.3)

We now want to give a property which will be satisfied by our representation ρ_{0} and which implies (3.3).

Proposition 3.4. Assume that for any s ∈ S the following three conditions are satisfied:

(i) All the entries of the matrix of ρ_{0}(s) in the basis e_{r} are non-negative.

(ii) ρ_{0}(s)(e_{s}) = 0, (iii) For any r ∈ R\{s} corresponding to a positive root β, the coefficient of ρ_{0}(s)(e_{r}) on

a basis element e_{r′} is not zero if and only if the reflection r′ corresponds to a root of the form s(β) − nα with n ∈ ℕ, where α is the simple root corresponding to s.

Then (3.3) is satisfied.

Proof: As ρ_{0}(s) has non-negative coefficients, condition (3.3) is equivalent to the two following conditions for any closed subset A of R:

(a) For any r ∈ R\A the coefficient of ρ_{0}(s)(e_{r}) on e_{r′} is not zero only if r′ ∈ R\s.A.

(b) For any r′ ∈ R\s.A there exists r ∈ R\A such that the coefficient of ρ_{0}(s)(e_{r}) on e_{r′} is not zero. Let us show that (a) and (b) are implied by assumptions (i), (ii) and (iii). Let r be

in R\A, corresponding to a positive root β ∈ Φ^{+}\A_{Φ}. By assumption (ii) there can be a non-zero coefficient in ρ_{0}(s)(e_{r}) only when β ≠ α and by assumption (iii) the only non-zero coefficients are on basis vectors e_{r′} where r′ corresponds to some s(β) − nα. As β ∉ A_{Φ}, we have (s(β) − nα) + nα = s(β) ∉ s(A_{Φ}) so by 2.6 we have r′ ∉ s.A. We have proved that (a) holds.

We show (b). By 2.6, the set s.A is the set of reflections with respect to the roots in {α} ∪ {β ∈ Φ^{+} | ∀n ∈ ℕ, β + nα ∈ Φ^{+} ⇒ β + nα ∈ s(A_{Φ})}. We can rewrite this set as {α} ∪ {β ∈ Φ^{+} | ∀n ∈ ℕ, s(β) − nα ∈ Φ^{+} ⇒ s(β) − nα ∈ A_{Φ}}, so by putting γ = s(β) it is equal to {α} ∪ s({γ ∈ Φ^{+}\{α} | ∀n ∈ ℕ, γ − nα ∈ Φ^{+} ⇒ γ − nα ∈ A_{Φ}}). So the set R\s.A corresponds to the set of positive roots s({γ ∈ Φ^{+}\{α} | ∃n ∈ ℕ, γ − nα ∈ Φ^{+}\A_{Φ}}).

Let s(γ) be in this set, then there exists β = γ − nα ∈ Φ^{+}\A_{Φ} and by assumption (iii) ρ_{0}(s)(e_{r}) has a non-zero coefficient on e_{r′} where r corresponds to β and r′ to s(γ).

We now define ρ_{0}. We shall first define ρ_{0} over the ring ℤ[q] where q is an indeterminate and we shall show that for any specialization of q into a real number in ]0, 1[, the representation satisfies (3.3).

We first make the construction when all the roots have same length, i.e., if W is of type A_{n}, D_{n} or E_{n}.

Proposition 3.5. If for s ∈ S corresponding to the simple root α and r ∈ R corresponding to a positive root β, we put

ρ_{0}(s)(e_{r}) = \{ \begin{matrix}0 & if β = α \\ e_{r} & if β ≠ α and β ± α ∉ Φ \\ (1 − q)e_{r} + qe_{srs} & if β + α ∈ Φ \\ e_{srs} & if β − α ∈ Φ \end{matrix}

then the endomorphisms ρ_{0}(s) for s ∈ S satisfy the braid relations so that ρ_{0} extends to a representation of B^{+}.

Note that as W is of type A, D or E, we cannot have β + α and β − α both in Φ. Proof: We have to show that if s and s′ in S correspond to two distinct simple roots α and α′ respectively, then ρ_{0}(s) and ρ_{0}(s′) satisfy the same braid relation as s and s′. To prove this, we compute the image under each side of the braid relation of any e_{r} where r is a reflection corresponding to some positive root β. We have to study all possible cases for the relative positions of α, α′ and β. This is a computation in the subsystem of roots in the space spanned by α, α′ and β. This subsystem can be of rank 2 or 3. If α and α′ are in two different irreducible components of the subsystem then, up to the swapping of s and s′, ρ_{0}(s′) acts trivially on e_{r} and on e_{srs} and the braid relation is true. So we can assume now that α and α′ are in the same component. If β is not in that component then both ρ_{0}(s) and ρ_{0}(s′) fix e_{r} and we are done. If β is equal to α or α′ or α + α′ then both sides of the braid relation map e_{r} to 0. We are left with the case where the subsystem is irreducible of rank 3, of type A_{3}. We leave to the reader the checking that in each one of the two possible cases, namely when the roots α and α′ are orthogonal (i.e., s and s′ commute) or when the roots α and α′ are adjacent in the Dynkin diagram, the two 6 by 6 sub-matrices satisfy the corresponding braid relation.

We now specialize q to some real number 0 < q < 1, and we still denote by ρ_{0} the specialized representation. With that choice, all entries of the matrix of ρ_{0}(b) for any b ∈ B^{+} are non-negative.

Proposition 3.6. The representation ρ_{0} satisfies (3.3).

Proof: We shall show that ρ_{0}(s) satisfies the assumptions of 3.4. The matrix of ρ_{0}(s) has only non-negative entries for any s ∈ S and the column indexed by s is zero. So (i) and (ii) in 3.4 are satisfied. As all roots have same length, the integer n in assumption (iii) of 3.4 can only be equal to 0 or 1. If β is a positive root distinct from α such that β ± α is not a root then ρ_{0}(e_{β}) has only one non-zero coefficient which is on e_{β} and s(β) = β. If β + α is a root then the only non-zero coefficients of ρ_{0}(s)(e_{β}) are on the basis elements corresponding to β = s(β) − α and β + α = s(β). If β − α is a root the only non-zero coefficient of ρ_{0}(e_{β}) is on e_{β−α} and β − α = s(β), so that 3.4 (iii) is satisfied.

As a consequence we have:

Corollary 3.7. Any representation ρ of B on V such that for any s ∈ S the matrix of ρ(s) in the basis (e_{r})_{r∈R} has its entries in ℝ[t] and specializes to ρ_{0}(s) for t = 0 is faithful.

We now want to define ρ as in the above corollary. We still assume that B is of type A_{n}, D_{n} or E_{n}. The existence of ρ is given by the following result.

Theorem 3.8. Let λ be a non-zero real number; then there exists a unique representation ρ of B^{+} on V with coefficients in ℝ[t] with respect to the basis (e_{r}) which specializes into ρ_{0} for t = 0 and verifies

(i) For all s ∈ S the matrix ρ(s) − ρ_{0}(s) has all its rows equal to 0 except the row corresponding to e_{s}.

(ii) For any s ∈ S we have ρ(s)(e_{s}) = tλe_{s}.

Moreover this representation satisfies (iii) For any pair (s_{1}, s_{2}) of distinct elements of S we have ρ(s_{1})(e_{s_{2}}) = ρ_{0}(s_{1})(e_{s_{2}}). (iv) For any pair s_{1} ∈ S corresponding to a simple root α and any r ∈ R corresponding to

a positive root β such that β − α is a root we have ρ(s)(e_{r}) = e_{r} + tλ\frac{q − 1}{q}e_{s}.

(v) The determinant of ρ(s) is equal to tλ(−q)^{n(s)} where 2n(s) is the number of reflections r such that srs ≠ r.

If s corresponds to the simple root α then n(s) is the number of positive roots β such that α + β is a root. Proof: We are looking, for each s ∈ S, for the row indexed by s in the matrix of ρ(s). These rows should be such that the braid relations are satisfied. Note that if such matrices exist they have the predicted determinant as they are block diagonal except for one row which has tλ on the diagonal, and they have a 2 by 2 diagonal block of determinant −q for each pair of roots {β, β + α} with β ∈ Φ^{+} and α corresponding to s, all other diagonal coefficients being equal to 1.

Let α be the simple root corresponding to s ∈ S; we have to determine the off-diagonal entries in the row of ρ(s) corresponding to the basis vector e_{s}. We want these entries to be multiples of t. For any positive root β ≠ α corresponding to a reflection r, we shall denote the entry indexed by s and r in the matrix of ρ(s) by tx_{β}^{α} (resp. ty_{β}^{α}, resp. tz_{β}^{α}) if α + β is a root (resp. if β − α is a root, resp. if β ± α is not a root). Proving the first assertion of the theorem is equivalent to proving that the set of equations on these unknowns, given by conditions (i) and (ii) of the theorem, has a unique solution.

For any pair of simple roots {α, α′} we have to write the conditions on the sets of unknowns under which the braid relations are satisfied by the corresponding ρ(s) and ρ(s′). We shall get an equation for any positive root β and any such pair of simple roots. The equation depends only on the root subsystem in the subspace spanned by α, α′ and β and on the coefficients of β in that subsystem. Let us consider first the case when that subsystem has rank 2. Then we have two possibilities for α and α′: either α and α′ are orthogonal and β is one of these two simple roots, or α + α′ is a root and β is equal to one of these two roots or to α + α′. The computation shows that the braid relations for ρ(s) and ρ(s′) are satisfied if and only if in the first case x_{α′}^{α} = x_{α}^{α′} = 0 and in the second case x_{α}^{α′} = x_{α′}^{α} = 0 and y_{α+α′}^{α′} = y_{α+α′}^{α} = tλ\frac{q − 1}{q}. In particular properties (iii) and (iv) will be satisfied by the solution if it exists.

We now consider the case when the subsystem of rank 3. The case when α, α′ and β are pairwise orthogonal gives no new equation. Some computation in the other cases leads to the following set of equations (recall that we consider only root systems of type A, D, or E) corresponding each to some Dynkin diagram for the subsystem of rank 3. We use the notation\underbrace{•_{γ_{1}} \begin{matrix}− \\ − \\ − \\ − \end{matrix}⋯\begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{γ_{l}}}_{β}to express that β = γ_{1} + ⋯ + γ_{l}.

(1) If •_{α} \begin{matrix}− \\ − \\ − \\ − \end{matrix}\underbrace{• ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •_{α′}}_{β} then x_{β}^{α} = x_{β−α′}^{α}.

(2) If •_{α}  \underbrace{•_{α′} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •}_{β} then z_{β}^{α} = z_{β−α′}^{α}.

(3) If •_{α} \begin{matrix}− \\ − \\ − \\ − \end{matrix}\underbrace{•_{α′} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •}_{β} then x_{β}^{α} = \frac{1}{q}(x_{β−α′}^{α′} + (q − 1)z_{β−α′}^{α}).

(4) If \underbrace{•_{α′} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •_{α} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •}_{β} then z_{β}^{α} = \frac{1}{q}((q − 1)y_{β−α′}^{α} + z_{β−α−α′}^{α′}).

(5) If \underbrace{•_{α} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; • ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •_{α′}}_{β} then y_{β}^{α} = y_{β−α′}^{α}.

(6) If \underbrace{•_{α} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •_{α′} ; \begin{matrix}− \\ − \\ − \\ − \end{matrix} ; •}_{β} then y_{β}^{α} = y_{β−α}^{α′}.

(7) If •_{α} \begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{α_{1}} •_{β} then z_{β}^{α} = z_{β}^{α_{1}}.

It should be pointed out that, although we should get a priori more equations, the other equations we get are the same up to taking β − α′ for β in equations (2) and (3) and taking β − α − α′ for β in (4).

From this set of equations we deduce the unicity of the solution by induction on the height of the roots. For any β and any α, one of the equations (1) to (6) gives the unknowns relative to β and α in terms of unknowns relative to a root of height strictly less than β. If β has height one then the unknowns are given by the rank 2 case and, as shown above, the solution to the rank two case is expressed by properties (iv) and (v) of the theorem. This shows the unicity and gives an algorithm to compute a solution (see below). We have to show that the solution exists. This could be done by providing the solution and checking the equations (see the formulas in section 4). We want give a direct proof.

As the equations are homogeneous it follows that multiplying λ by a non-zero constant amounts to multiplying t by this constant. So without loss of generality we can take λ = q^{2} which simplifies the computations. We prove the existence by induction on the height of β. The starting point of the induction is for β of height one for the unknowns x_{β}^{α} and z_{β}^{α} and for β of height two for the unknowns y_{β}^{α}. We have already seen that condition (ii) and properties (iii) and (iv) in the theorem give the solution in these cases. Assume we have a solution for all roots of height less than β. Let us remark first that equations (5) and (6) give y_{α}^{β} = q(q − 1), using property (iv) of the theorem. for any β and α such that β − α is a root. So we now have to solve equations (1) to (4) and (7), in which we can replace all unknowns of type y by q(q −1). We use the following algorithm depending on an arbitrary choice of an ordering of the simple roots. For a given β we take α′ to be the first simple root such that β − α′ is a root. Then one of the equations (1) to (4) (and only one) gives the corresponding unknown.

We have to show that the values obtained satisfy equation (7) and that a different ordering (i.e., a different choice of α′) gives the same values. We first consider equation (7). There are three cases as an unknown of type z can be given by equation (2) or (4). If the equation relevant for α and α_{1} is equation (2) then we note that z_{β−α′}^{α} = z_{β−α′}^{α_{1}} by (7), by induction. If the equation relevant for α is equation (2) and equation relevant for α_{1}

is equation (4), then we have •_{α} \begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{α_{1}} \begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{α′}. So we get z_{β−α′}^{α} = (q − 1)^{2} + \frac{1}{q}z_{β−α−α_{1}−α′}^{α_{1}} =

(q − 1)^{2} + \frac{1}{q}z_{β−α−α_{1}−α′}^{α′} = (q − 1)^{2} + \frac{1}{q}z_{β−α′−α_{1}}^{α′} by induction and equations (4), (7) and (2) respectively, which gives the result. The last case, where the equation relevant to α and α_{1} would be equation (4) is impossible.

We now prove that a different choice of α′ gives the same value for the unknowns. We use the following straightforward lemma.

Lemma 3.9. Let us consider two simple roots α′ and α′′ in a root system of type A, D or E and a positive root β such that β − α′ and β − α′′ are roots. Then β − α′ − α′′ is either equal to 0 or is a root. In the latter case α′ and α′′ are orthogonal.

Let β be a positive root as above, let α′ and α′′ be simple roots such that β − α′ and β − α′′ are roots and let α be as above, corresponding to s ∈ S. We want to show by induction on the height of β that equations (1) to (4) with α′ replaced by α′′ give the same values for the unknowns x_{β}^{α} or z_{β}^{α}. As in the above proof for equation (7), we consider each possible case.

If β ⊥ α and if equation (2) is relevant both for α′ and α′′ then by 3.9 and induction we have z_{β−α′}^{α} = z_{β−α′−α′′}^{α} = z_{β−α′′}^{α} if β − α′ − α′′ is a root, otherwise β = α′ + α′′ and z_{β−α′}^{α} = 0 = z_{β−α′′}^{α}, whence the result in this case.

If β ⊥ α and equation (2) is relevant for α′ and equation (4) for α′′ then β ≠ α′ + α′′ so by 3.9 β − α′ − α′′ is a root and α′ ⊥ α′′. By induction, equation (4) can be applied

to give z_{β−α′}^{α} = (q − 1)^{2} + \frac{1}{q}z_{β−α−α′−α′′}^{α′′} and equation (2) gives z_{β−α−α′′}^{α′′} = z_{β−α−α′−α′′}^{α′′},

whence the result.

If β ⊥ α and equation (4) is relevant for both α′ and α′′, then necessarily we have β = α + α′ + α′′ and then z_{β−α−α′}^{α′} = 0 = z_{β−α−α′′}^{α′′}.

If β + α is a root and equation (1) is relevant for both α′ and α′′ then β ≠ α′ + α′′ and by 3.9 and induction we have x_{β−α′}^{α} = x_{β−α′−α′′}^{α} = x_{β−α′′}^{α}.

If β + α is a root and equation (1) is relevant for α′ and equation (3) for α′′ then if β = α′ + α′′ we get x_{β−α′}^{α} = 0 = x_{β−α′′}^{α′′} = z_{β−α′′}^{α}. Otherwise we apply again 3.9 and as we

have α′ ⊥ α and α′ ⊥ α′′, we get by induction x_{β−α′}^{α} = \frac{1}{q}(x_{β−α′−α′′}^{α′′} + (q − 1)z_{β−α′−α′′}^{α})

and x_{β−α′−α′′}^{α′′} = x_{β−α′′}^{α′′} and z_{β−α′−α′′}^{α} = z_{β−α′′}^{α}, which gives the result.

If β + α is a root and equation (3) is relevant for both α′ and α′′ then β ≠ α′ + α′′ and by 3.9 β − α′ − α′′ is a root. Computing the inner product of this root with α shows that β − α − α′ − α′′ is also a root. We then have by induction

\begin{aligned}x_{β−α′}^{α′} + (q − 1)z_{β−α′}^{α} & =^{(1)} x_{β−α′−α′′}^{α′} + (q − 1)z_{β−α′}^{α} \\ & =^{(3)} \frac{1}{q}(x_{β−α−α′−α′′}^{α} + (q − 1)z_{β−α−α′−α′′}^{α′}) + (q − 1)z_{β−α′}^{α} \end{aligned}

and symmetrically x_{β−α′′}^{α′′} + (q − 1)z_{β−α′′}^{α} = \frac{1}{q}(x_{β−α−α′−α′′}^{α} + (q − 1)z_{β−α−α′−α′′}^{α′′}) + (q −

1)z_{β−α′′}^{α} We have to see that \frac{1}{q}z_{β−α−α′−α′′}^{α′} + z_{β−α′}^{α} = \frac{1}{q}z_{β−α−α′−α′′}^{α′′} + z_{β−α′′}^{α}. But by (4)

we have \frac{1}{q}z_{β−α−α′−α′′}^{α′} − z_{β−α′′}^{α} = (q − 1)^{2} and similarly when α′ and α′′ are exchanged, whence the result in this case.

Corollary 3.10. The representation of B^{+} defined by 3.8 extends to a faithful representation of B.

Proof: We deduce that ρ extends to B from 3.8 (v), as ρ(s) is invertible for any s ∈ S. The corollary is then a straightforward consequence of 3.8 and of 3.7.

We now want to get a similar representation for any type of crystallographic Coxeter group. As we have said in the introduction we consider only the cases when W is irreducible. So we have to look at Artin groups of type C_{n}, G_{2} or F_{4}. Each of these groups is the group of fixed elements, in a group of type A_{2n−1}, D_{4} or E_{6} respectively, under a diagram automorphism. The unicity statement in theorem 3.8 proves the following:

Corollary 3.11. For any diagram automorphism σ, the automorphism of the representation space V, given by the corresponding permutation π_{σ} of the basis (e_{r}), extends the representation to the semi-direct product B ⋊ <σ>.

Using this result we get that the group B^{σ} acts on the subspace of fixed points under π_{σ} in V. We compute in each of the three cases the representation we get, taking the basis of V^{σ} consisting of the sums of the orbits of σ in the basis (e_{r}) of V. Such an orbit corresponds to a root in a root system with Weyl group W^{σ}. In each case we get a representation ρ which specializes for t = 0 to a representation of (B^{+})^{σ} satisfying the faithfulness criterion 3.4 and satisfying conditions (i) and (ii) of theorem 3.8. We give below in section 4 the explicit values in each case.

Note also that theorem 3.8 shows that the matrices giving ρ(s) are compatible with the restriction to a standard parabolic subgroup.

In the following sections we shall always take λ = q^{2} to get simpler formulas.

4. Formulas. We give here the matrices ρ(s) in some cases. We denote by s an element of S corresponding to the simple root α, by s its image in S and by r a reflection corresponding to a positive root β.

Type A_{n}. If B is of type A_{n} we have

ρ(s)(e_{r}) = \{ \begin{matrix}tq^{2}e_{r} & if r = s \\ qe_{srs} + (1 − q)e_{r} & if α + β is a root \\ e_{srs} + tq(q − 1)e_{s} & if β − α is a root \\ e_{r} & if β ⊥ α and if α is not in the support of β \\ e_{r} + t(q − 1)^{2}e_{s} & if β ⊥ α and β has α in its support \end{matrix}

Type D_{n}. If W is of type D_{n} the entries will depend on the coefficients of the positive roots on the simple roots. We denote by α_{1} and α_{2} the simple reflections which are swapped by the diagram automorphism. For any positive root β we denote by n(β) the number of coefficients equal to 2 in the decomposition of β. If α is a simple root we denote by c_{α}(β) the coefficient of β on α. We say that α is an end of β if c_{α}(β) = 1 and the support of β − α is connected (β − α is not necessarily a root). We say that α is the right end of β if it is an end different from α_{1} or α_{2}.

With the same notation as in the A_{n} case we have:

ρ(s)(e_{r}) = \{ \begin{matrix}tq^{2}e_{r} & if r = s \\ e_{srs} + tq(q − 1)e_{s} & if β − α ∈ Φ \\ qe_{srs} + (1 − q)e_{r} & if α + β ∈ Φ and n(α + β) = 0 \\ qe_{srs} + (1 − q)e_{r} + t\frac{(q − 1)^{2}(q^{i} − 1)}{q^{i}}e_{s} & if α + β ∈ Φ and n(α + β) = i ≥ 1 \\ e_{r} & if β ⊥ α et c_{α}(β) = 0 \\ e_{r} + t(q − 1)^{2}e_{s} & if β ⊥ α, c_{α}(β) = 1 \\ & and α is not an end of β \\ e_{r} + t\frac{(q − 1)(q^{i+1} − 1)}{q^{i}}e_{s} & if β ⊥ α, α is the right end of β \\ & and n(β) = i \\ e_{r} + t\frac{(q − 1)^{2}(q + 1)}{q}e_{s} & if β ⊥ α and \\ & c_{α}(β) = 2 or α = α_{1} or α_{2}. \end{matrix}

Type C_{n}. We now give the formulas for type C_{n}. The Coxeter diagram is•_{s_{1}} \begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{s_{2}}⋯ •_{s_{n−1}} = •_{s_{n}}.

We make the choice of considering it as the Weyl group of a root sytem of type C_{n} so that the root α_{n} corresponding to s_{n} is a long root. We give first the value of ρ_{0}(s) for s ∈ S corresponding to the simple root α. For any maximal set of positive roots of the form {β, β + α, …, β + nα}, we give the sub-matrix of ρ_{0}(s) on the corresponding subspace. There are 3 cases as n can be equal to 0,1 or 2.

(i) If n = 0 the matrix is ( 1 ).

(ii) If n = 1 the matrix is ( \begin{matrix}1 − q & 1 \\ q & 0 \end{matrix} ).

(iii) If n = 2 the matrix is ( \begin{matrix}(1 − q)^{2} & 2(1 − q) & 1 \\ q(1 − q) & q & 0 \\ q^{2} & 0 & 0 \end{matrix} ).

We see on these formulas that the faithfulness criterion is satisfied under its form 3.4. We now give the row indexed by s in ρ(s). Let r ∈ R correspond to the positive root

β. If β = α the coefficient of ρ(s)(e_{s}) on e_{s} is tq^{2}. We assume in the following that β ≠ α.

For s ≠ s_{n},

(i) If β − α is a root the coefficient of ρ(s)(e_{r}) on e_{s} is equal to tq(q − 1) if β − α has coefficient 0 on α or if β − 2α is a root and is equal to t(q − 1)(2q − 1) otherwise.

(ii) If β + α is a root and β − α is not a root the coefficient is 0 unless β has a non-zero

coefficient on α in which case it is equal to t(q − 1)^{2}. (iii) If β ± α is not a root the coefficient is 0 if β has coefficient 0 on α, is 2t(q − 1)^{2} if β

has coefficient 2 on α and β is a short root and is t(q − 1)^{2} otherwise.

We now give the same information for s = s_{n}.

(i) If β − α is a root then the coefficient is 2tq(q − 1). (ii) If β + α is a root then the coefficient is 0. (iii) If β ± α is not a root then the coefficient is 0 if β has a zero coefficient on α, otherwise

it is t(q − 1)^{2} if β is a long root and 2t(q − 1)^{2} if β is a short root. Type G_{2}. We now give the representation, again denoted by ρ, for a group of type G_{2}. Let s_{1} and s_{2} be the two generators. We assume that s_{1} corresponds to the short simple root α_{1} and s_{2} corresponds to the long simple root α_{2}. We order the roots (and the reflections) as α_{1}, α_{2}, α_{2} + α_{1}, α_{2} + 2α_{1}, α_{2} + 3α_{1}, 2α_{2} + 3α_{1}. The matrices of s_{1} and s_{2} are then

ρ(s_{1}) = ( \begin{matrix}tq^{2} & & tq(q − 1) & 2tq(q − 1) & tq(q − 1) & \frac{t(q − 1)(q^{2} − 1)}{q} \\ & (1 − q)^{3} & 3(1 − q)^{2} & 3(1 − q) & 1 \\ & q(1 − q)^{2} & 2q(1 − q) & q \\ & q^{2}(1 − q) & q^{2} \\ & q^{3} \\ & & & & & 1 \end{matrix} )

ρ(s_{2}) = ( \begin{matrix}1 − q & & 1 \\ & tq^{2} & 3tq(q − 1) & 3t(1 − q)^{2} & \frac{t(q − 1)^{3}}{q} & tq(q − 1) \\ q \\ & & & 1 \\ & & & & 1 − q & 1 \\ & & & & q \end{matrix} )

These formulas show that the representation is faithful by criterion 3.4. Type F_{4}. In the same way one gets the representation for type F_{4} from the representation for type E_{6}. The matrices ρ_{0}(s) are given by the same formulas as for type C_{n}, so that here again criterion 3.4 is satisfied, We give in the appendix the row corresponding to e_{s} in ρ(s) for each s ∈ S.

5. Complements and remarks. We do not know in general if the above representations are irreducible. In the A_{n} case this has been proved by Zinno [Z]. But we can prove the following, which has already been proved by Krammer for type A_{n}. Proposition 5.1. For any Artin group of crystallographic type, the representation ρ which we have constructed in section 3 has no other endomorphisms than the scalar multiplications. Proof: Let f be an endomorphism of ρ. For any s ∈ S the linear map ρ(s) has a one dimensional eigenspace for the eigenvalue tq^{2}, so that f must map the line spanned by e_{s} on itself for any s ∈ S. We put f(e_{s}) = λ_{s}e_{s}. We first show that λ_{s} is independent of s ∈ S.

Lemma 5.2. Assume that s is a simple reflection corresponding to the simple root α, that r is a reflection corresponding to a positive root β such that β + α is a root and β + 2α is not a root; let r′ be the reflection corresponding to α + β. We assume moreover that f(e_{r}) = λe_{r} for some scalar λ. We denote by tx_{β}^{α} the coefficient of ρ(s)(e_{r}) on e_{s}. We have:

(i) If either x_{β}^{α} = 0 or λ = λ_{s} then f(e_{r′}) = λe_{r′}. (ii) If f(e_{r′}) = λ′e_{r′} for some scalar λ′ then λ = λ′ = λ_{s}.

Proof: From ρ(s)(f(e_{r})) = f(ρ(s)(e_{r})) we get a(1 − q)λe_{r} + qλe_{r′} + λtx_{β}^{α}e_{s} = λa(1 − q)e_{r} + qf(e_{r′}) + λ_{s}tx_{β}^{α}e_{s} for some a = 1, 2 or 3 as can be seen from the formulas for ρ(s). So, if either x_{β}^{α} = 0 or λ = λ_{s} we deduce f(e_{r′}) = λe_{r′}.

Conversely, with the same notation, if f(e_{r′}) = λ′e_{r′} and f(e_{r}) = λe_{r}, from the equality ρ(s)(f(e_{r′})) = f(ρ(s)(e_{r′})) we get λ′e_{r} + tyλ′e_{s} = λe_{r} + tyλ_{s}e_{s}, for some non-zero scalar y, whence λ = λ′ = λ_{s}. The fact that y ≠ 0 can be seen e.g. in the formulas of section 4 and the appendix.

Let α and α_{1} be two simple roots adjacent in the Dynkin diagram, corresponding to s and s_{1} in S; then α + α_{1} is a root and α + 2α_{1} is not a root, up to exchanging α and α_{1}. Moreover x_{α}^{α_{1}} = x_{α_{1}}^{α} = 0. So we can apply 5.2, (i) to get f(e_{r′}) = λ_{s_{1}}e_{r′} where r′ is the reflection corresponding to α + α_{1}. Then by 5.2, (ii) we get λ_{s} = λ_{s_{1}}.

So, as the Dynkin diagram is connected we have shown that λ_{s} is independent of s. Let us denote by λ this common value. Applying again 5.2 (i) we get that f(e_{r}) = λe_{r} for any reflection r in a parabolic subgroup of type A_{n}, D_{n} or E_{n}. In particular the proposition is proved for types A, D, E.

It remains to prove the proposition for types C_{n}, F_{4} or G_{2}. We have to show that if λ above is equal to 1 then f = Id. This is a straightforward consequence of the following lemma.

Lemma 5.3. If s ∈ S, corresponding to the simple root α, is such that f(e_{s}) = e_{s}, if r is a reflection corresponding to a positive root β such that β − α is not a root, that β + α, …, β + nα are positive roots and that β + (n + 1)α is not a root, if f(e_{r}) = e_{r}, then f fixes e_{r_{i}} for i = 1 … n, where r_{i} is the reflection corresponding to β + iα.

Proof: We get this result with similar arguments to the proof of 5.2, using the antitriangular shape of the sub-matrix of ρ(s) indexed by α, β + α, β + 2α, … β + nα.

As an application of the above proposition we can get for each Artin group of crystallographic type, the matrix of ρ(Δ) where Δ is the longest element of B_{red}. We know that this element is central in types C_{n}, D_{2n}, F_{4}, G_{2}, E_{7} and E_{8}. If Δ is not central its action by conjugation on B is the diagram automorphism of B. This diagram automorphism can be realized in the representation by π_{σ} as in 3.11, so in all cases, by proposition 5.1, we know ρ(Δ) up to a scalar. We know the determinant of ρ(Δ) as we know the determinant of ρ(s) for any s ∈ S: for types A, D, E, this determinant is tq^{2}(−q)^{n(s)} where n(s) is the number of positive roots which can be added to the simple root corresponding to s; for the other cases it is easily computable from the formulas in section 4. So we know ρ(Δ) up to a root of unity. We now use the following fact: is we specialize q to 1 in the representation we get a representation ρ_{q=1} where each s ∈ S acts by a monomial matrix having all its coefficients but one equal to 1 the last one being equal to t. From this we deduce that the non-zero coefficients in ρ_{q=1}(Δ) are equal to t, so that ρ(Δ) = tπ_{σ} times a power of q. This power is for example q^{n+1} for type A_{n}, or q^{2n} for type C_{n}, or q^{2n−2} for type D_{n}, or q^{6} for G_{2}, or q^{12} for E_{6}. Note that we could also have said that if σ is a diagram automorphism of W then the set (B_{red})^{σ} is equal to the set of “reduced” elements of B^{σ} so that we get ρ(Δ) for a group of type B_{n}, F_{4} or G_{2} by restriction from a group of type A, D or E.

We can also get a similar result to Krammer’s [K, 6.1] by showing that there exists a matrix T which does not involve t and conjugates ρ(s) to ρ(s)_{1/q}^{−1} where ρ(s)_{1/q} is the matrix we get by changing q into q^{−1} in ρ(s). From this it is possible to show along the same lines as in [K] that the greatest power of t which appears in ρ(b) for b ∈ B^{+} is equal to the number of terms in the normal form of b. It can also be shown that b ∈ B^{+} is divisible by Δ if and only if all the coefficients of ρ(b) are (polynomials) divisible by t.

6. Appendix.

Type F_{4}. We give here for an Artin group of type F_{4} the coefficients of t in the row corresponding to e_{s} in ρ(s) for each s ∈ S. We denote by s_{1}, s_{2}, s_{3}, s_{4} the simple reflexions in such a way that the Dynkin diagram is•_{s_{1}} \begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{s_{2}}\begin{matrix}< \\ = \end{matrix}•_{s_{3}} \begin{matrix}− \\ − \\ − \\ − \end{matrix} •_{s_{4}}with s_{1} and s_{2} corresponding to short roots and s_{3} and s_{4} to long roots. If we denote each root by the list of its coefficients on the simple roots, we order the positive roots as

[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1], [1, 1, 1, 0],
[0, 1, 2, 0], [0, 1, 1, 1], [1, 1, 2, 0], [1, 1, 1, 1], [0, 1, 2, 1], [1, 2, 2, 0], [1, 1, 2, 1], [0, 1, 2, 2],
[1, 2, 2, 1], [1, 1, 2, 2], [1, 2, 3, 1], [1, 2, 2, 2], [1, 2, 3, 2], [1, 2, 4, 2], [1, 3, 4, 2], [2, 3, 4, 2].

We then have: First row of ρ(s_{1}) divided by t:

q^{2}, 0, 0, 0, q(q −1), 0, 0, q(q −1), 0, 0, q(q −1), q(q −1), 0, q(q −1), q(q −1), 0, \frac{(q^{3} − 1)(q − 1)}{q^{2}},

q(q−1), \frac{(q^{3} − 1)(q − 1)}{q^{2}}, q(q−1), \frac{(q^{3} − 1)(q − 1)}{q^{2}}, q(q−1), \frac{(q^{3} − 1)(q − 1)}{q^{2}}, \frac{(q − 1)(2q^{3} − 1)}{q^{2}}

Second row of ρ(s_{2}) divided by t:

0, q^{2}, 0, 0, q(q − 1), q(q − 1), 0, (q − 1)^{2}, q(q − 1), q(q − 1), (q − 1)^{2}, (q − 1)^{2}, \frac{(q^{2} − 1)(q − 1)}{q},

(2q − 1)(q − 1), \frac{(q^{2} − 1)(q − 1)^{2}}{q^{2}} , q(q − 1), q(q − 1), (q − 1)^{2}, \frac{(q^{3} − 1)(q − 1)}{q^{2}}, (2q − 1)(q − 1),

\frac{(q^{3} − 1)(q − 1)}{q^{2}}, \frac{(q^{2} − 1)(2q − 1)(q − 1)}{q^{2}}, \frac{(q − 1)(2q^{2} − 1)}{q}, \frac{(q − 1)(2q^{3} − q^{2} − q + 1)}{q^{2}}

Third row of ρ(s_{3}) divided by t:

0, 0, q^{2}, 0, 0, (q−1)^{2}, q(q−1), (q−1)^{2}, 2q(q−1), \frac{(q − 1)^{3}}{q} , 2q(q−1), \frac{(q − 1)^{3}}{q} , q(q−1), 2(q−1)^{2},

q(q − 1), 2(q − 1)^{2}, \frac{(q^{2} − 1)^{2}(q − 1)}{q^{3}}, 2(q − 1)^{2}, q(q − 1), \frac{2(q − 1)^{3}}{q} , \frac{(q^{3} − 1)(q − 1)}{q^{2}},

2q(q − 1), \frac{2(q^{2} − 1)(q − 1)}{q}, \frac{2(q^{2} − 1)(q − 1)}{q}

Fourth row of ρ(s_{4}) divided by t:

0, 0, 0, q^{2}, 0, 0, q(q −1), 0, 0, q(q −1), 0, q(q −1), \frac{(q^{2} − 1)(q − 1)}{q}, 0, \frac{(q^{2} − 1)(q − 1)}{q}, 2q(q −1),

\frac{(q^{2} − 1)(q − 1)}{q}, 2q(q − 1), \frac{(q^{3} − 1)(q^{2} − 1)(q − 1)}{q^{4}}, 2q(q − 1), q(q − 1), \frac{2(q^{2} − 1)(q − 1)}{q},

\frac{2(q^{2} − 1)(q − 1)}{q}, \frac{2(q^{2} − 1)(q − 1)}{q}

Acknowledgments.

The first draft of this paper was written during a stay in the CIRM in Marseille-Luminy which I thank for providing a suitable environment for working. I also thank Jean-Yves Hée for the numerous and relevant remarks he made on a previous version.

7. References.

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[D] P. Deligne, Les immeubles des groupes de tresses généralisés, Inventiones math. 17, (1972) 273–302.

[Dy] M. Dyer, Reflection subgroups of Coxeter systems. J. Algebra 135 (1990), no. 1, 57–73.

[K] D. Krammer, Braid Groups are Linear preprint (2000)

[Mi] J. Michel, A note on words in braid monoids, J. of Algebra, 215 (1999) 366–377.

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[Z] M. Zinno, On Krammer’s representation of the braid group, preprint (2000).

LAMFA (CNRS), Faculté de mathématiques et informatique Université de Picardie-Jules Verne 33 Rue Saint Leu 80039 Amiens France

digne@u-picardie.fr