Root systems, symmetries and linear representations of Artin groups

BLAISE PASCAL

Olivier Geneste, Jean-Yves Hée & Luis Paris

Root systems, symmetries and linear representations of Artin groups Volume 26, n^o1 (2019), p. 25-54.

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Root systems, symmetries and linear representations of Artin groups

Olivier Geneste Jean-Yves Hée

Luis Paris

Abstract

LetΓbe a Coxeter graph, letWbe its associated Coxeter group, and letGbe a group of symmetries ofΓ. Recall that, by a theorem of Hée and Mühlherr,W^Gis a Coxeter group associated to some Coxeter graphbΓ. We denote byΦ⁺the set of positive roots ofΓand byΦb⁺the set of positive roots ofbΓ. Let Ebe a vector space over a fieldKhaving a basis in one-to-one correspondence withΦ⁺. The action ofGonΓinduces an action ofGonΦ⁺, and therefore onE. We show thatE^Gcontains a linearly independent family of vectors naturally in one-to-one correspondence withbΦ⁺and we determine exactly when this family is a basis ofE^G. This question is motivated by the construction of Krammer’s style linear representations for non simply laced Artin groups.

Systèmes de racines, symétries et représentations linéaires des groupes d’Artin

Résumé

SoientΓun graphe de Coxeter,Wson groupe de Coxeter associé etGun groupe de symétries deΓ.

Rappelons que, par un théorème de Hée et Mühlherr,W^Gest un groupe de Coxeter associé à un certain graphe de CoxeterbΓ. On noteΦ⁺l’ensemble des racines positives deΓetΦb⁺l’ensemble des racines positives debΓ. SoitEun espace vectoriel sur un corpsKayant une base en bijection avecΦ⁺. L’action de GsurΓinduit une action deGsurΦ⁺, et donc surE. Nous montrons queE^Gcontient une famille libre de vecteurs naturellement en bijection avecbΦ⁺et nous déterminons exactement quand cette famille est une base deE^G. Cette question est motivée par la construction de représentations linéaires à la Krammer de groupes d’Artin non simplement lacés.

1. Introduction

1.1. Motivation

Bigelow [1] and Krammer [20] proved that the braid groups are linear answering a historical question in the subject. More precisely, they proved that some linear representation ψ : B_n → GL(E)of the braid groupB_n previously introduced by Lawrence [22] is faithful. A useful information for us is thatEis a vector space over the fieldK=Q(q,z) of rational functions in two variables q,zoverQ, and has a natural basis of the form {e_i,j|1≤i< j ≤n}.

Keywords:Artin group, linear representation, Coxeter group, root system.

2010Mathematics Subject Classification:20F36.

LetΓbe a Coxeter graph, letWΓbe its associated Coxeter group, letAΓbe its associated Artin group, and letA⁺_Γbe its associated Artin monoid. The Coxeter graphΓis called of spherical typeifWΓis finite, it is calledsimply lacedif none of its edges is labelled, and it is calledtriangle freeif there are no three vertices inΓtwo by two connected by edges.

Shortly after the release of the papers by Bigelow [1] and Krammer [20], Digne [12]

and independently Cohen–Wales [6] extended Krammer’s [20] constructions and proofs to the Artin groups associated with simply laced Coxeter graphs of spherical type, and, afterwards, Paris [25] extended them to all the Artin groups associated to simply laced triangle free Coxeter graphs (see also Hée [16] for a simplified proof of the faithfulness of the representation). More precisely, for a finite simply laced triangle free Coxeter graphΓ, they constructed a linear representationψ:AΓ →GL(E), they showed that this representation is always faithful on the Artin monoidA⁺_Γ, and they showed that it is faithful on the whole groupAΓifΓis of spherical type. What is important to know here is thatE is still a vector space overK=Q(q,z)and thatEhas a natural basisB={e_β|β∈Φ⁺} in one-to-one correspondence with the setΦ⁺of positive roots ofΓ.

The question that motivated the beginning of the present study is to find a way to extend the construction of this linear representation to other Artin groups, or, at least, to some Artin groups whose Coxeter graphs are not simply laced and triangle free. A first approach would be to extend Paris’ [25] construction to other Coxeter graphs that are not simply laced and triangle free. Unfortunately, explicit calculations on simple examples convinced us that this approach does not work.

However, an idea for constructing such linear representations for some Artin groups associated to non simply laced Coxeter graphs can be found in Digne [12]. In that paper Digne takes a Coxeter graph Γ of type A_2n+1, D_n, orE₆ and consider some specific symmetrygofΓ. By Hée [14] and Mühlherr [24] the subgroupW_Γ^gof fixed elements by gis itself a Coxeter group associated with a precise Coxeter graphbΓ. By Michel [23], Crisp [7, 8] and Dehornoy–Paris [10], the subgroupA^g_ΓofAΓof fixed elements bygis an Artin group associated withbΓ. On the other hand the symmetrygacts on the basis BofEand the linear representationψ : A_Γ →GL(E)is equivariant in the sense that ψ(g(a)) =gψ(a)g⁻¹ for alla ∈ AΓ. It follows thatψ induces a linear representation ψ^g : A

bΓ → GL(E^g), whereE^g denotes the subspace of fixed vectors ofE under the action ofg. Then Digne [12] proves thatψ^gis faithful and thatE^ghas a “natural” basis in one-to-one correspondence with the setΦb⁺of positive roots ofbΓ. This defines a linear representation for the Artin groups associated with the Coxeter graphsBn(n≥2),G2

andF₄.

LetΓbe a finite simply laced triangle free Coxeter graph and letGbe a non-trivial group of symmetries ofΓ. ThenGacts on the groupsWΓandAΓand on the monoidA⁺_Γ.

We know by Hée [14] and Mühlherr [24] (see also Crisp [7, 8], Geneste–Paris [13] and Theorem 2.5) thatW_Γ^Gis the Coxeter group associated with some precise Coxeter graph bΓ. Moreover, by Crisp [7, 8], the monoidA^+G_Γ is an Artin monoid associated withbΓand in many cases the group A^G_Γ is an Artin group associated withbΓ. On the other hand,G acts on the basisBofE, and the linear representationψ :AΓ →GL(E)is equivariant in the sense thatψ(g(a))=gψ(a)g⁻¹for alla ∈ AΓ and allg ∈G. Thus,ψinduces a linear representationψ^G :A

bΓ→GL(E^G), whereE^G ={x∈E|g(x)=xfor allg∈G}.

We also know by Castella [3, 5] that the induced representationψ^G: A

bΓ→GL(E^G)is faithful on the monoidA⁺

bΓ. So, it remains to determine whenE^Ghas a “natural” basis in one-to-one correspondence with the setΦb⁺of positive roots ofbΓ. The purpose of this paper is to answer this question.

1.2. Statements

The simply laced triangle free Artin groups and the linear representationsψ:AΓ→GL(E) form the framework of our motivation, but they are not needed for the rest of the paper.

We will also work with any Coxeter graph, which may have labels and infinitely many vertices. So, letΓbe a Coxeter graph associated with a Coxeter matrixM=(m_s,t)_s,t∈S, letKbe a field, and letEbe a vector space overKhaving a basisB={e_β|β∈Φ⁺}in one-to-one correspondence with the setΦ⁺of positive roots ofΓ.

AsymmetryofΓis defined to be a permutationgofSsatisfyingm_g(s),g(t)=m_s,tfor all s,t∈S. The group of symmetries ofΓwill be denoted by Sym(Γ). LetGbe a subgroup of Sym(Γ). Again, we know by Hée [14] and Mühlherr [24] thatW_Γ^Gis the Coxeter group associated with some Coxeter graphbΓ. On the other hand,Gacts on the setΦ⁺of positive roots of Γand therefore onE. LetbΦ⁺ be the set of positive roots ofbΓ. In this paper we show thatE^G contains a “natural” linearly independent setBb={be

βb|bβ ∈Φb⁺} in one-to-one correspondence with the setΦb⁺and we determine whenBbis a basis ofE^G. From now on we will say that the pair(Γ,G)has thebΦ⁺-basis propertyif the above mentioned subsetBbis a basis ofE^G.

We proceed in three steps to determine the pairs(Γ,G)that have theΦb⁺-basis property.

In a first step (see Subsection 4.1) we show that it suffices to consider the case where all the orbits ofSunder the action ofGare finite. LetS_findenote the union of the finite orbits ofSunder the action ofG, and letΓ_findenote the full subgraph ofΓspanned byS_fin. Each symmetryg ∈GstabilizesS_fin, hence induces a symmetry ofΓ_fin. We denote byG_fin the subgroup of Sym(Γfin)of all these symmetries. In Subsection 4.1 we will prove the following.

Theorem 1.1. The pair(Γ,G)has thebΦ⁺-basis property if and only if the pair(Γ_fin,G_fin) has thebΦ⁺-basis property.

By Theorem 1.1 we can assume that all the orbits ofSunder the action ofGare finite.

In a second step (see Subsection 4.2) we show that it suffices to consider the case whereΓ is connected. LetΓ_i,i∈I, be the connected components ofΓ. For eachi∈Iwe denote by StabG(Γ_i)the stabilizer ofΓ_iinG. Each symmetryg∈StabG(Γ_i)induces a symmetry of Γ_i. We denote byG_i the subgroup of Sym(Γ_i)of all these symmetries. In Subsection 4.2 we will prove the following.

Theorem 1.2. Suppose that all the orbits ofSunder the action ofGare finite. Then the pair(Γ,G)has thebΦ⁺-basis property if and only if for eachi∈Ithe pair(Γ_i,G_i)has the Φb⁺-basis property.

By Theorem 1.2 we can also assume that Γ is connected. In a third step (see Subsection 4.3 and Subsection 4.4) we determine all the pairs (Γ,G) that have the Φb⁺-basis property withΓconnected and all the orbits ofSunder the action ofGbeing finite.

One can associate with Γ a real vector space V = É

s∈SRα_s whose basis is in one-to-one correspondence withSand acanonical bilinear formh ·,· i :V×V →R. These objects will be defined in Subsection 2.1. We say thatΓis ofspherical typeifSis finite andh ·,· iis positive definite, and we say thatΓis ofaffine typeifSis finite and h ·,· iis positive but not positive definite. A classification of the connected spherical and affine type Coxeter graphs can be found in Bourbaki [2]. In this paper we use the notations A_m(m≥1),. . . ,I₂(p)(p=5 orp≥7) of Bourbaki [2, Chap. VI, §4, No 1, Thm. 1] for the connected Coxeter graphs of spherical type, and the notationsAe₁,Ae_m(m≥2),. . . , Ge₂of Bourbaki [2, Chap. VI, §4, No 2, Thm. 4] for the connected Coxeter graphs of affine type. Moreover, we use the same numbering of the vertices of these Coxeter graphs as the one in Bourbaki [2, Planches] with the convention that the unnumbered vertex in Bourbaki [2, Planches] is here labelled with 0.

To the Coxeter graphs of spherical type and affine type we must add the two infinite Coxeter graphs∞A∞andD∞drawn in Figure 1.1. The Coxeter graphA∞of the figure does not appear in the statement of Theorem 1.3 but it will appear in its proof. These Coxeter graphs are part of the family of so-calledlocally spherical Coxeter graphsstudied by Hée [15, Texte 10].

Now, the conclusion of the third step which is in some sense the main result of the paper is the following theorem, proved in Subsection 4.4.

1 2 n

1 2

3 4 n

A∞ D∞

-n -1 0 1 n

∞A∞

Figure 1.1. Locally spherical Coxeter graphs

Theorem 1.3. Suppose thatΓis connected,Gis non-trivial and all the orbits ofSunder the action ofGare finite. Then(Γ,G)has thebΦ⁺-basis property if and only if(Γ,G)is up to isomorphism one of the following pairs (see Figure 1.2, Figure 1.3 and Figure 1.4).

(i) Γ=A_2m+1(m≥1) andG=hgiwhere

g(s_i)=s_2m+2−i (1≤i ≤2m+1). (ii) Γ=D_m(m≥4) andG=hgiwhere

g(s_i)=si(1≤i≤m−2), g(s_m−1)=sm, g(s_m)=sm−1. (iii) Γ=D₄andhg₁i ⊂G⊂ hg₁,g₂iwhere

g₁(s₁)=s₃, g₁(s₂)=s₂, g₁(s₃)=s₄, g₁(s₄)=s₁, g₂(s₁)=s₁, g₂(s₂)=s₂, g₂(s₃)=s₄, g₂(s₄)=s₃. (iv) Γ=E₆andG=hgiwhere

g(s₁)=s6, g(s₂)=s2, g(s₃)=s5, g(s₄)=s4, g(s₅)=s3, g(s₆)=s1. (v) Γ=eA_2m+1(m≥1) andG=hgiwhere

g(s₀)=s0, g(s_i)=s_2m+2−i (1≤i≤2m+1). (vi) Γ=De_m(m≥4) andG=hgiwhere

g(s_i)=si (0≤i≤m−2), g(s_m−1)=sm, g(s_m)=sm−1. (vii) Γ=De₄andhg₁i ⊂G⊂ hg₁,g₂iwhere

g₁(s₀)=s₀, g₁(s₁)=s₃, g₁(s₂)=s₂, g₁(s₃)=s₄, g₁(s₄)=s₁, g₂(s₀)=s₀, g₂(s₁)=s₁, g₂(s₂)=s₂, g₂(s₃)=s₄, g₂(s₄)=s₃.

(viii) Γ=Ee₆andG=hgiwhere

g(s₀)=s0, g(s₁)=s6, g(s₂)=s2, g(s₃)=s5, g(s₄)=s4, g(s₅)=s3, g(s₆)=s1. (ix) Γ=∞A∞andG=hgiwhere

g(s_i)=s_−i (i∈Z). (x) Γ=D∞andG=hgiwhere

g(s₁)=s₂, g(s₂)=s₁, g(s_i)=s_i (i≥3).

1 2 m

m+1 2m m+2

2m+1

g 1 2 m-3 m-2

m-1

m g

(i) (A_2m+1,hgi) (ii) (D_m,hgi) 1 2

3

4 g1

g₂ 2 4

3 1

5 6

g

(iii) (D₄,hg₁,g₂i) (iv) (E₆,hgi)

Figure 1.2. Pairs with theΦb⁺-basis property: spherical type cases

1.3. Linear representations

We return to our initial motivation before starting the proofs. Recall that a Coxeter graph Γis calledsimply lacedifm_s,t ∈ {2,3}for alls,t∈S,s,t, and thatΓis calledtriangle freeif there are no three distinct verticess,t,r∈Ssuch thatm_s,t,m_t,r,m_r,s ≥3. Suppose thatΓis a finite, simply laced and triangle free Coxeter graph. LetAΓbe the Artin group and A⁺_Γ be the Artin monoid associated withΓ. Suppose thatK= Q(q,z)and E is a vector space overKhaving a basisB ={e_α|α ∈Φ⁺}in one-to-one correspondence with the setΦ⁺of positive roots ofΓ. By Krammer [20], Cohen–Wales [6], Digne [12]

and Paris [25], there is a linear representationψ : AΓ →GL(E)which is faithful ifΓ is of spherical type and which is always faithful on the monoid A⁺_Γ. LetGbe a group of symmetries ofΓ. Recall from Subsection 1.1 thatW_Γ^Gis a Coxeter group associated with a precise Coxeter graphbΓand by Crisp [7, 8] we have(A⁺_Γ)^G =A⁺

bΓ. ThenGacts

0

1 2 m

m+1 m+2 2m+1

g

0 1

2 3 m-3 m-2

m-1

m g

(v) (eA_2m+1,hgi) (vi) (De_m,hgi) 1

2 0

3

4 g₁

g₂ 0 2 4

3 1

5 6

g

(vii) (De₄,hg₁,g₂i) (viii) (Ee₆,hgi)

Figure 1.3. Pairs with thebΦ⁺-basis property: affine type cases

0

1 2 n

-1 -2 -n

g

1

2

3 4 n

g

(i x) (∞A∞,hgi) (x) (D∞,hgi)

Figure 1.4. Pairs with theΦb⁺-basis property: locally spherical type cases

onE, the linear representationψis equivariant, and it induces a linear representation ψ^G:A

bΓ →GL(E^G). By Castella [3, 5] this representation is always faithful onA⁺

bΓand is faithful on the whole A

bΓifΓis of spherical type.

One can find an explicit description ofbΓin Crisp [7, 8] and in Geneste–Paris [13]. In particular, we havebΓ=B_m+1in Case (i) of Theorem 1.3, we havebΓ=Bm−1in Case (ii), bΓ=G₂ in Case (iii),bΓ =F₄in Case (iv),bΓ=Ce_m+1 ifm≥2 andbΓ =Be₂ifm=1 in Case (v),bΓ=Be_m−1in Case (vi),bΓ=Ge₂in Case (vii), andbΓ=Fe₄in Case (viii).

So, concerning a description of a linear representationψ : AΓ →GL(E)as above, whereE has a given basisB={e_α|α ∈Φ⁺}in one-to-one correspondence with the setΦ⁺of positive roots ofΓ, forΓof spherical or affine type, the situation is as follows.

For the following Coxeter graphs of spherical type the construction is done and the representation is faithful on the whole group AΓ.

• A_m, (m≥1): original work of Krammer [20].

• D_m (m ≥ 4), E₆, E₇, E₈: due to Digne [12] and independently to Cohen–

Wales [6].

• B_m(m≥2),F₄, G₂: due to Digne [12].

Such representations for the Coxeter graphs H₃,H₄ and I₂(p)(p = 5 orp ≥ 7) are unknown. For the following Coxeter graphs of affine type the construction is done and the representation is faithful on the Artin monoid A⁺_Γ.

• Ae_m(m≥3),De_m(m≥4),Ee₆,Ee₇,Ee₈: due to Paris [25].

• Ae₂: due to Castella [4].

• Be_m(m≥2),Ce_m(m≥3),Fe₄,Ge₂: due Castella [3, 5] for the construction of the representationψ :AΓ→GL(E)and for the proof of the faithfulness ofψonA⁺_Γ, and due to the present work for an explicit construction of a basis in one-to-one correspondence withΦ⁺.

Curiously a construction for the remaining Coxeter graph of affine type,Ae₁, is unknown.

1.4. Organization of the paper

The paper is organized as follows. In Subsection 2.1 and Subsection 2.2 we give preliminaries on root systems and symmetries. In Section 3 we define the subsetBb= {be

βb|bβ ∈ bΦ⁺}ofE^G. Section 4 is dedicated to the proofs. Theorem 1.1 is proved in Subsection 4.1 and Theorem 1.2 is proved in Subsection 4.2. The proof of Theorem 1.3 is divided into two parts. In a first part (see Subsection 4.3) we show that, under the assumptions “finite orbits andΓconnected”, thebΦ⁺-basis property is quite restrictive.

More precisely we prove the following.

Proposition 1.4. Suppose thatΓis a connected Coxeter graph,Gis a non-trivial group of symmetries ofΓ, and all the orbits ofSunder the action ofGare finite. If(Γ,G)has thebΦ⁺-basis property, thenΓis one of the following Coxeter graphs:A_2m+1(m≥1),D_m (m≥4),E₆,Ae_2m+1(m≥1),De_m(m≥4),Ee₆,Ee₇,∞A∞,D∞.

In a second part (see Subsection 4.4) we study all the possible pairs(Γ,G)withΓin the list of Proposition 1.4 andGnon-trivial to prove Theorem 1.3.

2. Preliminaries

2.1. Root systems

LetSbe a (finite or infinite) set. ACoxeter matrixonSis a square matrixM =(m_s,t)_s,t∈S indexed by the elements ofSwith coefficients inN∪ {∞}, such thatm_s,s=1 for alls∈S, andms,t=mt,s≥2 for alls,t∈S,s,t. This matrix is represented by itsCoxeter graph, Γ, defined as follows. The set of vertices ofΓisS, two verticess,tare connected by an edge ifm_s,t ≥3, and this edge is labelled withm_s,tifm_s,t ≥4.

TheCoxeter groupassociated withΓis the groupW =WΓdefined by the presentation W =hS|s² =1 for alls∈S, (st)^ms,t =1 for alls,t∈S, s,tandm_s,t,∞i. The pair(W,S)is called theCoxeter systemassociated withΓ.

There are several notions of “root systems” attached to all Coxeter groups. The most commonly used is that defined by Deodhar [11] and taken up by Humphreys [17]. As Geneste–Paris [13] pointed out, this definition is not suitable for studying symmetries of Coxeter graphs. In that case it is better to take the more general definition given by Krammer [19, 21] and taken up by Davis [9], or the even more general one given by Hée [14]. We will use the latter in this paper.

We callroot prebasisa quadrupleB=(S,V,Π,R), whereSis a (finite or infinite) set, Π={α_s|s∈S}is a set in one-to-one correspondence withS,V =É

s∈SRα_sis a real vector space withΠas a basis, andR={σ_s|s∈S}is a collection of linear reflections of Vsuch thatσ_s(α_s)=−α_sfor alls∈S. For alls,t ∈Swe denote byms,tthe order ofσ_sσ_t. ThenM=(m_s,t)_s,t∈Sis a Coxeter matrix. The Coxeter group of this matrix is denoted by W(B). We have a linear representation f :W(B) →GL(V)which sendsstoσ_sfor all s∈S. Since we do not need to specify the map f in general, forw∈Wandx∈V, the vector f(w)(x)will be simply writtenw(x). We setΦ(B)={w(α_s) |s∈Sandw ∈W} and we denote byΦ(B)⁺(resp.Φ(B)⁻) the set of elementsβ ∈ Φ(B)that are written β=Í

s∈Sλ_sα_swithλ_s≥0 (resp.λ_s ≤0) for alls∈S. We say thatBis aroot basisif we have the disjoint unionΦ(B)=Φ(B)⁺tΦ(B)⁻. In that caseΦ(B)is called aroot system and the elements ofΦ(B)⁺(resp.Φ(B)⁻) are calledpositive roots(resp.negative roots).

Finally, we say thatBis areduced root basisand thatΦ(B)is areduced root systemif, in addition, we haveRα_s∩Φ(B)={α_s,−α_s}for alls∈S.

Example. LetΓbe a Coxeter graph associated with a Coxeter matrixM =(m_s,t)_s,t∈S. As before, we setΠ={α_s|s∈S}andV =É

s∈SRα_s. We define a symmetric bilinear formh ·,· i:V×V →Rby

hα_s, α_ti=

(−2 cos(π/m_s,t) ifms,t,∞,

−2 ifm_s,t=∞.

For eachs∈Swe define a reflectionσ_s :V →V byσ_s(x)=x− hα_s,xiα_sand we set R={σ_s|s∈S}. By Deodhar [11],B=(S,V,Π,R)is a root basis and, by Bourbaki [2], WΓ=W(B)and the representation f :WΓ→GL(V)is faithful. This root basis is reduced since we havehβ, βi=2 for allβ∈Φ(B). It is called thecanonical root basisofΓand the bilinear formh ·,· iis called thecanonical bilinear formofΓ.

In this subsection we give the results on root systems that we will need, and we refer to Hée [14] for the proofs and more results.

Theorem 2.1(Hée [14]). LetB=(S,V,Π,R)be a root basis. Then the induced linear representation f :W(B) →GL(V)is faithful.

Let(W,S)be a Coxeter system. The word length of an elementw∈W with respect to Swill be denoted by lg(w)=lg_S(w). The set ofreflectionsof(W,S)is defined to beR= {wsw⁻¹|w∈Wands∈S}. The following explains why the basisB={e_β|β∈Φ⁺}of our vector spaceE will depend only on the Coxeter graph (or on the Coxeter system) and not on the root system.

Theorem 2.2(Hée [14]). LetB=(S,V,Π,R)be a reduced root basis, letW =W(B), and letΦ=Φ(B). LetRbe the set of reflections of(W,S). Letβ∈Φ. Letw ∈W and s∈Ssuch thatw(α_s)=β. We haveβ∈Φ⁺if and only iflg(ws)>lg(w). In that case the element$(β)=wsw⁻¹ ∈Rdoes not depend on the choice ofwands. Moreover, the map$:Φ⁺→Rdefined in this way is a one-to-one correspondence.

LetΓbe a Coxeter graph and let(W,S)be its associated Coxeter system. ForX ⊂S we denote byΓ_Xthe full subgraph ofΓspanned byX and byWX the subgroup ofW generated byX. By Bourbaki [2],(W_X,X)is the Coxeter system ofΓ_X. IfB=(S,V,Π,R) is a reduced root basis, then we denote by V_X the vector subspace ofV spanned by Π_X={α_s|s∈X}and we setR_X={σ_s|_VX|s∈X}.

Proposition 2.3(Hée [14]). LetB=(S,V,Π,R)be a reduced root basis and letX ⊂S.

LetW =W(B). Thenσ_s(V_X)=VXfor alls∈X, the quadrupleBX=(X,VX,Π_X,R_X)is a reduced root basis,Φ(BX)=Φ(B) ∩V_X, andW_X=W(BX).

The following is proved in Bourbaki [2] and is crucial in many works on Coxeter groups. It is important to us as well.

Proposition 2.4(Bourbaki [2]). The following conditions on an elementw₀ ∈W are equivalent.

(i) For allu∈W we havelg(w0)=lg(u)+lg(u⁻¹w₀).

(ii) For alls∈Swe havelg(sw₀)<lg(w₀).

Moreover, such an elementw₀exists if and onlyW is finite. Ifw₀satisfies(i)and/or(ii) thenw₀is unique,w₀is involutive (i.e.w₀²=1), andw₀Sw₀=S.

WhenWis finite the elementw₀in Proposition 2.4 is calledthe longest elementofW.

2.2. Symmetries

Recall that a symmetryof a Coxeter graphΓ associated with a Coxeter matrix M = (m_s,t)_s,t∈Sis a permutationgofSsatisfyingm_g(s),g(t)=m_s,tfor alls,t ∈S. Recall also that Sym(Γ)denotes the group of all symmetries ofΓ. LetGbe a subgroup of Sym(Γ).

We denote byOthe set of orbits ofSunder the action ofG. Two subsets ofOwill play a special role. First, the setO_finconsisting of finite orbits. Then, the subsetS ⊂ O_finformed by the orbits X ∈ O_finsuch thatWX is finite. ForX ∈ Swe denote byuX the longest element ofW_X(see Proposition 2.4) and, forX,Y ∈ S, we denote bymb_X,Y the order of uXuY. Note thatMb=(mbX,Y)_X,Y∈Sis a Coxeter matrix. Its Coxeter graph is denoted by bΓ=bΓ^G. Finally, we denote byW^Gthe subgroup ofW of fixed elements under the action ofG.

Theorem 2.5(Hée [14], Mühlherr [24]). LetΓbe a Coxeter graph, letGbe a group of symmetries ofΓ, and let(W,S)be the Coxeter system associated withΓ. ThenW^Gis generated byS_W ={u_X|X ∈ S}and(W^G,S_W)is a Coxeter system associated withbΓ.

Take a root basisB=(S,V,Π,R)such thatW =W(B). The action ofGonSinduces an action ofGonVdefined byg(α_s)=α_g(s)for alls∈Sandg ∈G. We say thatBis symmetricwith respect toGifσ_g(s) =gσ_sg⁻¹ for alls ∈S andg ∈G. Note that the canonical root basis is symmetric whatever isG. Suppose thatBis symmetric with respect toG. Then the linear representation f :W → GL(V)associated withBis equivariant in the sense that f(g(w))=g f(w)g⁻¹ for allw∈W andg∈G. So, it induces a linear representation f^G :W^G→GL(V^G), whereV^G={x∈V|g(x)=xfor allg∈G}.

For eachX ∈ Swe setbα_X =Í

s∈Xα_sand we denote byVbthe vector subspace ofV spanned by bΠ={bα_X|X ∈ S}. We haveVb ⊂V^G but we have no equality in general.

However, we have f^G(w)(bV)=Vbfor allw∈W^G. We setbσ_X= f^G(u_X)|

Vbfor allX ∈ S andRb={bσ_X|X∈ S}.

Theorem 2.6(Hée [14]). The quadruplebB=(S,V,bΠ,b R)b is a root basis which is reduced ifBis reduced. Moreover, we haveW(bB)=W^G.

The root basisbBof Theorem 2.6 will be called theequivariant root basisofB/G.

3. Definition ofBb

From now onΓdenotes a given Coxeter graph,M =(m_s,t)_s,t∈Sdenotes its Coxeter matrix, and(W,S)denotes its Coxeter system. We take a group Gof symmetries ofΓ and a reduced root basisB=(S,V,Π,R)such thatW =W(B). We assume thatBis symmetric with respect toGand we use again the notations of Subsection 2.2 (bΓ,Mb=(mbX,Y)_X,Y∈S, W^G,bB=(S,V,bΠ,b R), and so on). We setb Φ=Φ(B)andΦb=Φ(bB).

LetKbe a field. We denote byEthe vector space overKhaving a basisB={e_α|α∈ Φ⁺}in one-to-one correspondence with the setΦ⁺of positive roots. The groupGacts on Evia its action onΦ⁺and we denote byE^Gthe vector subspace ofEof fixed vectors under this action. We denote byΩthe set of orbits and byΩ_finthe set of finite orbits of Φ⁺under the action ofG. For eachω∈Ω_finwe setbe_ω=Í

α∈ωe_α. It is easily shown that B₀ ={be_ω|ω∈Ω_fin}is a basis ofE^G.

The definition ofBbrequires the following two lemmas. The proof of the first one is left to the reader.

Lemma 3.1. SetV⁺ =Í

s∈SR+α_s. Let x,y ∈ V⁺ and s ∈ S. If x+y ∈ R+α_s, then x,y∈R+α_s.

For X ∈ Swe set ω_X = {α_s|s ∈ X}. Note that ω_X ∈ Ω_fin for allX ∈ S. More generally, we havew(ω_X) ∈Ω_finfor allX ∈ Sand allw∈W^G.

Lemma 3.2. LetX,X⁰∈ Sandw,w⁰∈W^G. Ifw(bα_X)=w⁰(bα_X⁰), thenw(ω_X)=w⁰(ω_X⁰).

Proof. Up to replacing the pair(w,w⁰)by(w⁰⁻¹w,1)we can assume thatw⁰=1. Then we havew(bα_X)=bα_X⁰and we must show thatw(ω_X)=ω_X⁰. For that it suffices to show that the intersection of the two orbitsw(ω_X)andω_X⁰is non-empty. Either all the roots w(α_s),s∈X, lie inΦ⁺, or all of them lie inΦ⁻. Moreover, their sumw(bα_X)=bα_X⁰lies in V_X⁺0. So, they all lie inΦ⁺_X0. Similarly, sincew⁻¹(bα_X⁰)=bα_X, all the rootsw⁻¹(α_t),t∈ X⁰, lie inΦ⁺_X. Lets ∈ X. We havew(α_s)=Í

t∈X⁰λ_tα_twithλ_t ≥0 for allt ∈ X⁰. Hence, we have α_s = Í

t∈X⁰λ_tw⁻¹(α_t)and all the vectorsλ_tw⁻¹(α_t),t ∈ X⁰, lie inV⁺. By Lemma 3.1 it follows that all these vectors lie inR+α_s. But the family{w⁻¹(α_t) |t∈ X⁰} is linearly independent, hence only one λ_tis nonzero. Thus, there existst ∈ X⁰such thatλ_t >0 andw(α_s)=λ_tα_t. Since the root basisBis reduced, we haveλ_t=1, hence w(α_s)=α_t ∈w(ω_X) ∩ω_X⁰, which completes the proof.

Now we can define a mapF=F_B,G:Φb⁺→Ω_finas follows. Letbα∈bΦ⁺. LetX ∈ S and w ∈ W^G such thatbα =w(bα_X). Then we set F(bα) = w(ω_X). By Lemma 3.2 the definition of this map does not depend on the choices ofwandX. Moreover, it is easily shown that it is injective. Now, we set

Bb={be_F(

bα)|bα∈bΦ⁺},

and we say that the pair(Γ,G)has theΦb⁺-basis propertyifBbis a basis ofE^G. Note that(Γ,G)has thebΦ⁺-basis property if and only ifBb=B₀. Equivalently,(Γ,G)has the Φb⁺-basis property if and only ifFis a bijection (or a surjection).

4. Proofs

4.1. Proof of Theorem 1.1

In this subsection we denote byS_finthe union of the finite orbits ofSunder the action ofG, and we setΓ_fin=Γ_Sfin,Vfin=VSfin,Π_fin=Π_SfinandR_fin=R_Sfin. We consider the root basisBfin =BSfin =(S_fin,V_fin,Π_fin,R_fin)and its root systemΦ_fin =Φ_Sfin =Φ(Bfin).

Each symmetryg ∈Ginduces a symmetry ofΓ_fin. We denote byGfinthe subgroup of Sym(Γ_fin)of all these symmetries.

Lemma 4.1. Letα∈Φ. The following assertions are equivalent.

(1) The orbitω(α)ofαunder the action ofGis finite.

(2) The rootαlies inΦ_fin.

Proof. Suppose that the orbitω(α)is finite. Thesupportof a vectorx=Í

s∈Sλ_sα_s∈V is defined to be Supp(x)={s∈S|λ_s,0}. The unionX_αof the supports of the roots β∈ω(α)is a finite set and is stable under the action ofG, henceX_αis a union of finite orbits. This implies thatXα⊂Sfin, henceα∈Φ∩Vfin, and therefore, by Proposition 2.3, α∈Φ_fin.

Suppose α ∈ Φ_fin. There existt₁,t₂, . . . ,t_n,s ∈ S_finsuch that α = (t₁t₂· · ·t_n)(α_s).

For eachg ∈ Gwe haveg(α) =(g(t₁)g(t₂) · · ·g(t_n))(α_g(s)). The respective orbits of t₁,t₂, . . . ,t_nandsare finite, hence the orbitω(α)={g(α) |g ∈G}is finite.

Proof of Theorem 1.1. We denote byΩ⁰the set of orbits ofΦ⁺

finunder the action ofGfin. On the other hand we denote bybBfinthe equivariant root basis ofBfin/G_fin. By Lemma 4.1 each orbit ofΦ⁺_finunder the action ofG_finis finite and each finite orbit inΦ⁺is contained inΦ⁺

fin, henceΩ⁰=Ω_fin. Moreover, since each elementX ∈ Sis contained inS_finand W^Gis generated byS_W ={u_X|X ∈ S}(see Theorem 2.5), we have(W_Sfin)^Gfin =W^G andbBfin=bB. So, we haveΦ(bBfin)⁺=Φ(bB)⁺andF_B,G =F_Bfin,Gfin:Φ(bB)⁺→Ω_fin. Since we know that(Γ,G)has thebΦ⁺-basis property if and only ifF_B,Gis a bijection,(Γ,G) has theΦb⁺-basis property if and only if(Γ_fin,G_fin)has thebΦ⁺-basis property.

4.2. Proof of Theorem 1.2

From now on we assume that all the orbits ofSunder the action ofGare finite. Then, by Lemma 4.1, the orbits ofΦunder the action ofGare also finite.

Lemma 4.2. We assume that for each rootα∈Φ⁺there exists∈Sandw∈W^Gsuch thatα=w(α_s). Lets,t ∈Sandg ∈Gsuch thatt =g(s),s. Thenm_s,t=2.

Proof. Suppose instead thatm_s,t ≥ 3. Set α = s(α_t). We haveα =α_t+λα_s where λ > 0. In particularα ∈ Φ⁺. So, there existw ∈W^G andr ∈ Ssuch thatw(α)=α_r. Sincet =g(s), we haveα_t =g(α_s), hencew(α_t)=w(g(α_s))=g(w(α_s)). This shows that either both roots w(α_s) andw(α_t)lie in Φ⁺, or they both lie in Φ⁻. Moreover, w(α)=w(α_t)+λw(α_s)=α_r, hence the two rootsw(α_s)andw(α_t)lie inΦ⁺. Thus, by Lemma 3.1, the two vectorsw(α_t)andλw(α_s)lie inR+α_r, which is a contradiction since, tbeing different froms, these two vectors are linearly independent.

Proposition 4.3. The following conditions are equivalent.

(1) The pair(Γ,G)has theΦb⁺-basis property.

(2) For each rootα∈Φ⁺there existw∈W^Gands∈Ssuch thatα=w(α_s).

(3) For each rootα∈Φthere existw∈W^Gands∈Ssuch thatα=w(α_s).

Proof. Suppose that(Γ,G)has thebΦ⁺-basis property. Letα∈Φ⁺. The orbitω(α)lies inΩ_fin=Ωand the mapF :Φb⁺ →Ω_finis a bijection, hence there existw ∈W^Gand X ∈ Ssuch thatw(ω_X)=ω(α). In particular, there exists an elements∈ X ⊂Ssuch thatw(α_s)=α.

Suppose that for eachα ∈Φ⁺there existw ∈W^G ands ∈S such thatα=w(α_s).

Letω ∈Ω_fin=Ω. Letα∈ω. By assumption there existw∈W^Gands∈ Ssuch that α=w(α_s). LetXbe the orbit ofsunder the action ofG. The setXis finite since it is an orbit and, by Lemma 4.2,WXis the direct product of|X|copies ofZ/2Z. So,WXis finite andX ∈ S. Setβb=w(bα_X). We haveβb∈bΦ⁺and the orbitF(β)b =w(ω_X)contains the rootα=w(α_s), hence it is equal toω. So, the mapFis a surjection, hence(Γ,G)has the Φb⁺-basis property.

Suppose that for eachα∈Φ⁺there existw∈W^Gands∈Ssuch thatα=w(α_s). In order to show that for eachα∈Φthere existw∈W^Gands∈Ssuch thatα=w(α_s), it suffices to consider a rootα∈Φ⁻. By assumption, since−α∈Φ⁺, there existw⁰∈W^G ands∈Ssuch that−α=w⁰(α_s). LetXbe the orbit ofs. By Lemma 4.2 the Coxeter graph Γ_Xis a finite union of isolated vertices, henceWXis finite,u_X =Î

t∈Xt, andu_X(α_t)=−α_t for allt∈X. Setw=w⁰uX. Thenw∈W^Gandw(α_s)=w⁰uX(α_s)=w⁰(−α_s)=α.

The implication (3)⇒(2) is obvious.

By combining Lemma 4.2 and Proposition 4.3 we get the following.

Lemma 4.4. Suppose that(Γ,G)has thebΦ⁺-basis property. LetXbe an orbit ofSunder the action ofG. Then Γ_X is a finite union of isolated vertices. In particular,X ∈ S, u_X =Î

t∈Xt, andu_X(α_t)=−α_tfor allt ∈X.

Proof of Theorem 1.2. LetΓ_i,i ∈ I, be the connected components ofΓ. Fori ∈ Iwe denote by Si the set of vertices ofΓ_i, and we setVi =VSi,Π_i =Π_Si, andR_i =R_Si. Consider the root basisBi =BSi =(S_i,V_i,Π_i,R_i)and its root systemΦ_i =Φ_Si =Φ(Bi).

Note thatΦis the disjoint union of theΦ_i’s andW is the direct sum of theWSi’s. Recall that each symmetryg ∈ StabG(Γ_i)induces a symmetry ofΓ_i and thatG_i denotes the subgroup of Sym(Γi)of these symmetries.

Suppose that(Γ,G)has thebΦ⁺-basis property. Leti∈I. Letα∈Φ_i. By Proposition 4.3 there exists ∈ S andw ∈ W^G such thatα =w(α_s). SinceW is the direct sum of the WSj’s,w is uniquely writtenw =Î

j∈Iw_j, wherew_j ∈ WSj for all j ∈ I, and there are only finitely many j ∈ I such thatw_j , 1. Let j ∈ I such thats ∈ S_j. We have α=w(α_s)=w_j(α_s) ∈Φ_j, hencei= jands∈S_i. On the other hand, ifg ∈StabG(Γ_i), theng(w)=wandg(W_Si)=W_Si, henceg(w_i)=w_i. So,w_i ∈W_S^Gi

i andα=w_i(α_s). By Proposition 4.3 we conclude that(Γ_i,Gi)has thebΦ⁺-basis property.

Suppose that(Γ_i,G_i)has theΦb⁺-basis property for alli∈I. Letα∈Φ. Leti∈Isuch thatα∈Φ_i. By Proposition 4.3 there existw_i ∈W_S^Gi

i ands ∈Si such thatα=w_i(α_s).

The action ofGonSinduces an action ofGonI defined byg(Γ_i)=Γ_g(i), forg ∈G andi∈I. Since the orbits ofSunder the action ofGare finite, the orbits ofIunder the action ofGare also finite. LetJ⊂Ibe the orbit ofi. For eachj ∈Jwe chooseg ∈G such thatg(i)=jand we setw_j =g(w_i) ∈W_Sj. The fact thatw_i ∈W_S^Gi

i implies that the definition ofw_j does not depend on the choice ofg. Letw=Î

j∈Jw_j. Thenw ∈W^G andw(α_s)=w_i(α_s)=α. By Proposition 4.3 we conclude that(Γ,G)has theΦb⁺-basis

property.

4.3. Proof of Proposition 1.4

From now on we assume that Γ is connected, that G is nontrivial, and that all the orbits ofSunder the action ofGare finite. We also assume thatBis the canonical root basis, h ·,· i : V×V → Ris the canonical bilinear form ofΓ, as they are defined in Subsection 2.1, and thatΦ=Φ(B)is the so-calledcanonical root system.

Recall that for alls,t∈Swe havehα_s, α_ti=−2 cos(π/m_s,t). In particularhα_s, α_ti =2 ifs =t,hα_s, α_ti =0 ifms,t =2,hα_s, α_ti =−1 ifms,t =3, and−2≤ hα_s, α_ti <−1 if 4≤m_s,t≤ ∞. Let≡denote the equivalence relation onΦgenerated by the relation≡₁

defined by:α≡₁ β⇔ hα, βi < {0,1,−1}. Note that the relation≡₁ is reflexive (since hα, αi=2 for allα∈Φ) and symmetric.

The proof of Proposition 1.4 is divided into two parts. In a first part (see Corollary 4.13) we prove that, if≡has at least two equivalence classes, thenΓis one of the Coxeter graphs A_m(m≥1),D_m(m≥4),E_m(6≤m≤8),Ae_m(m≥2),De_m(m≥4),Ee_m(6≤m≤8), A∞,∞A∞,D∞. In a second part (see Proposition 4.17) we prove that, if(Γ,G)has the Φb⁺-basis property, then≡has at least two equivalence classes.

Forα∈Φwe denote byr_α :V→Vthe linear reflection defined byr_α(x)=x− hα,xiα.

Note that, ifs∈Sandw∈Ware such thatα=w(α_s), thenr_α =wsw⁻¹. In particular r_α ∈Wfor allα∈Φandr_αs =sfor alls∈S.

Lemma 4.5.

(1) We haveα≡₁ −αfor allα∈Φ.

(2) Letα, β∈Φsuch thatα≡β. Thenw(α) ≡w(β)for allw∈W.

(3) Letα, β∈Φsuch thatα≡β. Thenα≡r_α(β)andβ≡r_α(β).

Proof. Part (1) is true sincehα,−αi =−2<{0,1,−1}. Part (2) follows from the fact that eachw∈W preserves the bilinear formh ·,· i. Letα, β∈Φsuch thatα≡β. By Part (2) we haverα(α) ≡ rα(β). Butrα(α) =−αhence, by Part (1),α ≡rα(β). We also have

β≡r_α(β)sinceα≡β.

Lemma 4.6. Assume that α_s ≡ α_t for all s,t ∈ S. Then all the elements of Φ are equivalent modulo the relation≡.

Proof. It suffices to prove that for eachα∈Φthere existss∈ Ssuch thatα≡α_s. Let α∈Φ. There existt∈Sandw∈W such thatα=w(α_t). We argue by induction on the length of w. We can assume thatw ,1. Then we havew =sw⁰withs ∈ S,w⁰∈ W and lg(w⁰)<lg(w). Setβ=w⁰(α_t). By induction there existsu ∈ Ssuch thatβ≡α_u. We haveα =w(α_t)=(sw⁰)(α_t)=s(β)hence, by Lemma 4.5 (2),α ≡s(α_u). But, by assumption,α_u≡α_s, hence, by Lemma 4.5 (3),α_s≡s(α_u) ≡α.

Recall that thesupportof a vectorx=Í

s∈Sλ_sα_s∈V is Supp(x)={s∈S|λ_s,0}.

Lets,t∈S. Apathfromstotoflength`is a sequences<sub>0</sub>,s<sub>1</sub>, . . . ,s<sub>`</sub> inSof length`+1 such thats<sub>0</sub> =s,s<sub>`</sub> =tandm_si−1,si ≥3 for alli ∈ {1, . . . , `}. Thedistancebetweens andt, denoted byd(s,t), is the shortest length of a path fromstot. Then thedistance between an elements∈Sand a subsetX⊂Sisd(s,X)=min{d(s,t) |t∈ X}.

Lemma 4.7. Letα=Í

s∈Sλ_sα_s ∈Φ⁺,t ∈S\Supp(α)andt0∈Supp(α). Assume that d(t,Supp(α))=d(t,t₀)andλ_t0 >1. Thenα≡α_t.

Proof. We argue by induction ond=d(t,Supp(α)). There exists a patht₀,t₁, . . . ,t_d of lengthdfromt0tot. We first show thatα≡₁α_t1. We havehα, α_t1i=Í

s∈Sλ_shα_s, α_t1i.

For eachs∈Supp(α)we haveλ_s >0 andhα_s, α_t1i ≤0 (sinces,t₁), hencehα, α_t1i ≤ λ_t0hα_t0, α_t1i. Moreover, we haveλ_t0 >1 and hα_t0, α_t1i ≤ −1 (sincemt0,t1 ≥ 3), hence hα, α_t1i <−1, and thereforeα≡₁ α_t1. By Lemma 4.5 (3) we also haveα ≡α⁰, where α⁰=t₁(α).

Now we can assume thatd ≥2. We writeα⁰=Í

s∈Sλ_s⁰α_s. We haveα⁰=α−hα, α_t1iα_t1, hence Supp(α⁰)=Supp(α) ∪ {t₁},λ_t⁰₁ =−hα, α_t1i >1, andd(t,Supp(α⁰))=d(t,t₁)= d−1, therefore, by induction,α⁰ ≡α_t. Finally, we haveα ≡α⁰andα⁰ ≡ α_t, hence

α≡α_t.

Lemma 4.8. Assume that there ares,t∈Ssuch thatm_s,t ≥4. Then the relation≡has only one equivalence class.

Proof. We havehα_s, α_ti =−2 cos(π/m_s,t) ≤ −2 cos(π/4)=−√

2<−1, henceα_s ≡α_t. By Lemma 4.5 (3) we also have α_s ≡ α where α = s(α_t). Let u ∈ S \ {s,t}. By Lemma 4.6 it suffices to show that eitherα_u ≡α_sorα_u≡α_t. We can and do assume that d(u,s) ≤d(u,t)and we show thatα_u≡α_s. Setα=λ_sα_s+α_t. We have Supp(α)={s,t}, d(u,Supp(α))=d(u,s), andλ_s=−hα_s, α_ti >1, hence, by Lemma 4.7,α_u≡α. So, since

α_s ≡α, we haveα_u ≡α_s.

Lemma 4.9. IfScontains a subsetYsuch that (1) ∅,Y ,S, and

(2) for alls∈Y there existsα=Í

r∈Yλ_rα_r ∈Φ⁺

Ysuch thatα≡α_sandλ_r >1for allr∈Supp(α),

then≡has only one equivalence class.

Proof. Lets∈Yandt ∈S\Y. There exists a rootα=Í

r∈Yλ_rα_r ∈Φ⁺

Ysuch thatα≡α_s andλ_r >1 for allr ∈Supp(α). Then, by Lemma 4.7,α≡α_t. So, sinceα≡α_s, we have α_s ≡α_t. Lets,s⁰∈Y. SinceY ,S, we can taket ∈S\Y. By the above,α_s ≡α_tand α_s⁰ ≡α_t, henceα_s≡α_s⁰. Lett,t⁰∈S\Y. SinceY ,∅, we can takes∈Y. By the above, α_s ≡α_tandα_s≡α_t⁰, henceα_t ≡α_t⁰. We conclude by applying Lemma 4.6.

Lemma 4.10. Suppose thatΓis one of the following Coxeter graphs of affine type: eA_m (m≥2),Dem(m≥4),Ee6,Ee7,Ee8. For eachs∈Sthere exists a rootα=Í

r∈Sλ_rα_r ∈Φ⁺ such thathα_s, αi=−2andλ_r ≥2for allr∈S.

Proof. We number the elements ofSas in Bourbaki [2, Planches] with the convention that the unnumbered vertex in Bourbaki [2, Planches] is here labelled with 0. Letβdenote the greatest root ofΦ_S\{0}. Here is the value ofβaccording toΓ.