Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type

Journal of Algebra 556 (2020) 621–633 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type Matthieu Calvez a,∗, Bruno A. Cisneros de la Cruz b, María Cumplido c,d aInstituto de Matemáticas, Universidad de Valparaíso, Gran Bretaña 1111, Valparaíso 2340000, Chile bInstituto de Matemáticas de la UNAM, Oaxaca - Consejo Nacional de Humanidades Ciencias y Tecnologías, León No.2 altos, Oaxaca de Juárez, México cInstitut de Mathématiques de Bourgogne, UMR 5584, CNRS, Univ. Bourgogne Franche-Comté, 21000 Dijon, France dDepartment of Mathematics, Heriot-Watt University, Riccarton, EH14 4AS Edinburgh, Scotland, UK a r t i c l e i n f o a b s t r a c t Article history: Received 10 February 2019 Available online 9 April 2020 Communicated by Gunter Malle Keywords: Artin-Tits group Conjugacy stability Parabolic subgroups We give a complete classification of conjugacy stable parabolic subgroups of Artin-Tits groups of spherical type. This answers a question posed by Ivan Marin and generalizes a theorem obtained by Juan González-Meneses in the specific case of Artin braid groups. © 2020 Elsevier Inc. All rights reserved. *Corresponding author. E-mail addresses: calv[email protected] (M. Calvez), [email protected] (B.A. Cisneros de la Cruz), [email protected] (M. Cumplido). https://doi.org/10.1016/j.jalgebra.2020.03.017 0021-8693/© 2020 Elsevier Inc. All rights reserved. 622 M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 Fig. 1. Connected Coxeter graphs of spherical type with a specific enumeration of the vertices. 1. Introduction Let Sbe a finite set. A Coxeter matrix over Sis a symmetric square matrix M= (ms,t)s,t∈Sindexed by the elements of S, such that ms,s =1, and ms,t ∈{2, 3, 4, ..., ∞} for all s, t ∈S, s =t. Such a Coxeter matrix is usually represented by its Coxeter graph, denoted by Γ =Γ S=Γ(M). This is a labeled graph whose set of vertices is S, in which two distinct vertices sand tare connected by an edge if ms,t ⩾3; if in addition ms,t ⩾4, the corresponding edge wears the label ms,t. The Artin-Tits system of Γis the pair (A, S), where A =AΓis the group AΓ=SΠ(s, t;ms,t)=Π(t, s;ms, t) for all s, t ∈S, s =t, ms,t =∞, where, for m ⩾2, Π(a, b;m)=(ab)kif m=2k, (ab)kaif m=2k+1. The group AΓis called the Artin-Tits group of Γ; sometimes we shall also use the notation ASto refer to this group. If we add to the presentation of AΓthe relations s2=1, for every s ∈S, we obtain the Coxeter group associated to AΓ. When this group is finite we say that AΓhas spherical type. By extension, we say that Γ is of spherical type if AΓhas spherical type. AΓis called irreducible if the graph Γis connected and reducible otherwise. Notice that if Γ1, ··· , Γrare the connected components of Γ, then AΓ=AΓ1×···×AΓr. We recall Coxeter’s classification [5]of connected Coxeter graphs of spherical type (hence of irreducible Artin-Tits groups of spherical type) in Fig. 1. The name of the graph will be used to refer to the corresponding Artin-Tits group; for instance the Artin-Tits group of type E7is the Artin-Tits group of the graph E7. Let Xbe a subset of S. The standard parabolic subgroup associated to Xis the subgroup of AΓgenerated by Xand denoted by AX. Consider the subgraph ΓXof Γ =Γ Sgenerated by X(the set of vertices is Xand the edges are exactly the edges of ΓSwhich connect two vertices in X). It is known [13]that (AX, X)is the Artin-Tits M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 623 system of ΓX. A parabolic subgroup is a subgroup Pconjugate to some standard parabolic subgroup AX. Note that Pand AXare isomorphic; if AXis irreducible of spherical type, the type of Pis the name of the graph ΓXin Fig. 1. The flagship example of an Artin-Tits group of spherical type is the braid group on n strands Bn(n ⩾2) [2]. It is associated to the Coxeter graph An−1depicted in Fig. 1; the corresponding Coxeter group is the symmetric group Sn. We recall that each generator siis the crossing of the strands in the positions iand i +1. Let mand nbe two positive integers such that 2 ⩽m ⩽n. Considering only the m −1 first vertices of the graph An−1furnishes a fundamental example of a standard (irreducible) parabolic subgroup: the braid group Bmembedded in Bnby adding n −mstraight strands to any m-strand braid. It was shown in [11]that the above embedding Bm→B n(for 2 ⩽m <n) does not merge conjugacy classes, i.e. if two m-strand braids are conjugate in the n-strand braid group, they must already be conjugate as m-strand braids. Motivated by the latter result, Ivan Marin asked some years ago whether standard parabolic subgroups of irreducible Artin-Tits groups of spherical type are conjugacy stable. A (non-trivial) proper subgroup Hof a group Gis said to be conjugacy stable if any two elements of Hwhich are conjugated in Gmust be conjugated through an element of H; this is equivalent to saying that the conjugacy classes of Hdo not merge in G. It is an easy exercise to check that conjugacy stability is preserved under subgroup conjugation; therefore Marin’s question actually covers all parabolic subgroups of irreducible Artin-Tits groups of spherical type. Suppose now that ASis a reducible Artin-Tits group of spherical type, expressed as the direct product AS=AS1×...×ASr, where r>1and each ASiis non-trivial and irreducible. For a subset XS, we can consider Xi=X∩Si(i =1, ..., r) and decompose AXas a direct product of parabolic subgroups AX=AX1×...×AXr–notice that AXimight be trivial (when Xiis empty) or reducible. Since elements in distinct components of AScommute pairwise, AXis conjugacy stable in ASif and only if AXi is conjugacy stable in ASifor all i. In view of the above remarks, the following, which is our main result, allows to decide the conjugacy stability of any given parabolic subgroup of any Artin-Tits group of spherical type: Theorem 1. Let AΓ=ASbe an irreducible Artin-Tits group of spherical type and let ∅ =XS. (1) If AXis irreducible, AXis conjugacy stable in ASexcept in the following cases: (a) ASis of type E6, E7or E8and AXis of type D, (b) ASis of type E8and AXis of type E7, (c) ASis of type Dand AXis of type D2k, (d) ASis of type H4and AXis of type H3. (2) If AXis reducible, AXis not conjugacy stable in ASexcept in the following cases: 624 M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 (a) ASis of type Bn(n ⩾3) and AX=A{s1}×AZ, with Z⊂{s3, ..., sn}and AZ irreducible. (b) ASis of type F4. González-Meneses’ proof in the specific case of braids relies heavily on the identification between braids and mapping classes of punctured disks: Birman-Lubotzky- McCarthy’s Canonical Reduction Systems of mapping classes play a fundamental role. Although more combinatorial in spirit, our approach was inspired by González-Meneses’: instead of the Canonical Reduction System, we use the parabolic closure of elements of Artin-Tits groups of spherical type introduced recently in [7]; see Theorem 7. The first main tool we will use are ribbons. These objects are highly useful when conjugating parabolic subgroups and we introduce them in Section 2. The other main result consists in making depend conjugacy stability of standard parabolic subgroups on a special property that we will call Property . This property and its implications will be explained in Section 3. Finally, in Section 4we finish the proof of Theorem 1. 2. Garside elements and ribbons Given a group Gand g, x ∈G, we denote by xg=g−1xg the conjugate of xby g; this defines a right-action of Gon itself. In the same way, for g∈Gand a subset Hof G, we denote by Hgthe set of g-conjugates of elements of H. For the remainder of the present section, we fix an irreducible Artin-Tits group of spherical type AS. The monoid A+ Sconsisting of positive elements (which can be written as words on Swith only positive exponents) is a Garside monoid (see [4,8]): this involves, among other things, a fundamental or Garside element which we denote by ΔS(for X⊂S, the Garside element of AXwill be denoted by ΔX). Example. In the braid group on n +1 strands (Artin-Tits group of type An), the Garside element is s1(s2s1)(s3s2s1) ···(snsn−1···s1); it can be seen as a half-twist of the trivial braid on n + 1 strands. Although the paper builds on previous works which use in a crucial way the Garside structure of AS, our arguments do not directly involve this structure so we only record some useful properties of the Garside element ΔS. It is known that conjugation by ΔSis an involution and that SΔS=S. Moreover, ΔS is central if ASis of type A1, Bn, Dn(neven), E7, E8, F4, H3, H4or I2m(meven) [4,9]. Table 1synthesizes the conjugacy action by ΔSin the other irreducible cases. In each of the cases considered in Table 1, we call flip automorphism the inner automorphism s → sΔSof AS. For more information about the specific construction of ΔS, see [4]. We are now able to define ribbons. Note that the following definition is slightly different from the original definition of ribbon from [10]based upon [12]. M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 625 Table 1 Conjugation by the special Garside element ΔS. ASAn(n⩾2) Dn(nodd) E6I2m(modd) sΔS isn−i+1,1⩽i⩽n⎧ ⎪ ⎨ ⎪ ⎩ s2i=1 s1i=2 si3⩽i⩽n ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s6i=1 s2i=2 s5i=3 s4i=4 s3i=5 s1i=6 s(i+1) mod 2 Definition 2. [7, Definition 4.1] Let ASbe an Artin–Tits group of spherical type. Given X⊆Sand t ∈S, we define the positive element r(t, X)=Δ −1 XΔX∪{t} and we call it a ribbon. If moreover tis adjacent to Xin the Coxeter graph ΓS, we say that r(t, X)is an adjacent ribbon. Remark 3. Notice that r(t, X) conjugates Xto some subset Xof X∪{t}and X= XΔX∪{t}. The forthcoming proofs use a weak version of a result from [1]and [7]. The support of a positive element gof ASis defined as Supp(g)={s∈S, s appears in any positive word on Srepresenting g}. Lemma 4 ([7, Corollary 6.5]; [1, Lemma 21]). Let g, hbe positive elements of an Artin– Tits group ASof spherical type such that Supp(g) =YSand Supp(h) =ZS. If g and hare conjugate in AS, then Yand Zare also conjugate. Moreover, there are subsets Y=Y0, ..., Yn=Zof Sand adjacent ribbons r(ti, Yi−1)(i =1, ..., n) conjugating Yi−1 to Yi. 3. The Property  In this section we introduce our Property and we show its sufficiency for conjugacy stability, in the spherical case. In a second step, we show that Property holds in several cases. Definition 5. Let (AS, S)be an Artin-Tits system (of spherical type) and let ∅ =XS We say that (AX, AS)satisfies Property if for all Y1, Y2⊂X, and g∈ASsuch that Y1g=Y2, there exists h ∈AXsuch that sh=sgfor all s ∈Y1. Proposition 6. Let (AS, S)be an Artin-Tits system of spherical type and let ∅ =XS. If (AX, AS)has Property , then AXis conjugacy stable in AS. 626 M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 Before proceeding to the proof, we recall the important and recently defined notion of parabolic closure of elements of Artin-Tits groups of spherical type: Theorem 7 ([7, Section 7, Lemma 8.1]). Let (AS, S)be an Artin-Tits system of spherical type. For each a ∈AS, there is a unique minimal (with respect to inclusion) parabolic subgroup Paof ASwhich contains a; we call this subgroup the parabolic closure of a. Furthermore, for g∈AS, we have Pag=Pag. Proof of Proposition 6.Let a, b ∈AXand c ∈ASsatisfying ac=b. According to Theorem 7, we have that Pac=Pac=Pb. According to [6, Theorem 3], both subgroups Paand Pbof AXcan be standardized inside AX: i.e. there exist α, β∈AXand subsets Ya, Ybof Xsuch that Paα=AYaand Pbβ=AYb. Notice that Aα−1cβ Ya=AYb. By [10, Proposition 2.1.(3)] we can find u ∈ASwith Yau=Yband v∈AYb, such that α−1cβ =uv. By Property , we can find u∈AXsuch that su=sufor every s ∈Ya. It follows that suv=suv =sα−1cβ for all s ∈Ya; therefore for any element z∈AYa, we have zuv=zα−1cβ . Applying this to the particular element aα∈AYa, we obtain aαuv=aαα−1cβ =acβ =bβ. It follows that b =aαuvβ−1, and we note that αuvβ−1∈AX. Remark 8. Given an Artin-Tits system (AS, S)of spherical type, Property for the pair (AX, AS) implies that the automorphisms of AXinduced by conjugation by an element in the normalizer NAS(AX)are inner automorphisms of AX. Indeed, we know [10, Theorem 0.1] that NAS(AX) =AX·QZAS(AX)(where QZAS(AX) ={g∈AS, Xg=X}); Property then says that for g∈QZAS(AX), we can find h ∈AXsuch that xg=xh for all x ∈AXand the claim follows. Now, we will see that Property holds in some cases. Lemma 9. Let AXbe an Artin–Tits group of type An, E6or I2·m, modd, and let ΓX be its defining Coxeter graph. Let Y1, Y2⊂X, let ΓY1, ΓY2be the respective induced subgraphs of ΓXand let ψ:Γ Y1−→ ΓY2be an isomorphism of labeled graphs. Then there exists v∈AXsuch that ψ(y) =yv, for all y∈Y1. Proof. Suppose first that AXis of type I2·m. Write ¯si=s(i+1) mod 2, for i =1, 2. Then ψ(y) =yfor all y∈Y1, or ψ(y) =¯yfor all y∈Y1. It then suffices to take vto be the identity or ΔX, accordingly. Suppose that AXis of type An. The graph ΓY1is a disjoint union of path graphs (possibly with a single vertex) and the graph isomorphism ψ: ΓY1−→ ΓY2can be realized conjugating first by a product of ribbons (as in the example of Fig. 2) and then by a product of the Garside elements of some irreducible components of AY2. Now suppose that AXis of type E6. It can be checked that in this case, two standard parabolic subgroups ΓY1and ΓY2are isomorphic if and only if Y1and Y2are conjugate; therefore under our hypothesis Y1and Y2must be conjugate in AX. M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 627 Fig. 2. In the braid group B12, a braid made of ribbons conjugating Y1={s1, s2} ∪{s6} ∪{s9, s10, s11}to Y2={s2, s3, s4} ∪{s8} ∪{s10, s11}. Table 2 The conjugacy classes of subsets of Xwith no representative lying in {s1, s3, s4, s5, s6}. Y1Type of Y1Automorphism of ΓY1v {s1,s 2,s 4,s 6}A1×A1×A2 flip on A2-component transposition of the A1-components Δ{s2,s4} ΔX {s1,s 2,s 3,s 5,s 6}A1×A2×A2 flip on the first A2-component flip on the second A2-component transposition of the A2-components Δ{s1,s3} Δ{s5,s6} ΔX {s2,s 3,s 4,s 5}D4 transposition of the leaves s3,s 5of ΓY1 transposition of the leaves s2,s 5of ΓY1 ΔX Δ{s1,s2,s3,s4,s5} {s1,s 2,s 4,s 5,s 6}A1×A4flip on the A4-component Δ{s2,s4,s5,s6} XE 6flip of the whole ΓXΔX If Y1or Y2is a subset of {s1, s3, s4, s5, s6}, we are back to the previous case, as A{s1,s3,s4,s5,s6}is a braid group on 6 strands. In Table 2, we list the conjugacy classes of subsets of Xwith no representative in {s1, s3, s4, s5, s6}. In each case, we see that every possible graph automorphism of ΓY1(the table considers generators of the automorphism group of ΓY1) is induced by conjugation by some v∈AX, given explicitly in the last column.  Remark 10. Although this will not be used in the sequel, we note that the statement of Lemma 9holds as well for AXof type E8. It also is easily seen that the statement of Lemma 9is not true if AXis of type Bor D. If AXis of type B, it suffices to consider Y1=Y2={s1, s2}and the automorphism ψ of ΓY1which permutes its two vertices; it can be checked that s1and s2are not conjugate in AX(see Lemma 13), so ψfails to be induced by an inner automorphism of AX. If AX has type D, choosing Y1={s1, s2}and Y2={s1, s4}, we’ve just seen in the above proof that the graph isomorphism ΓY1−→ ΓY2given by s1→ s1and s2→ s4is not induced by any inner automorphism of AX. 628 M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 Proposition 11. Let ASbe any Artin–Tits group, let ∅ =XSsuch that AXis of type An, E6or I2·m, modd. Then (AX, AS)has Property . Proof. Let Y1, Y2⊂Xand w∈ASsuch that Yw 1=Y2; in particular, conjugation by w induces an isomorphism between the labeled graphs ΓY1and ΓY2and it follows from Lemma 9that we can find v∈AXso that yv=yw, for all y∈Y1. Proposition 12. Let ASbe an irreducible Artin–Tits group of spherical type and let ΓS be its defining Coxeter graph. Let ∅ =XS. Assume that: •AXis of type Bn, or •AXis of type Dn(nodd) and ASis of type Dm(m >n). Then (AX, AS)satisfies Property . Proof. Whenever Z⊂S, we write ΓZfor the subgraph of ΓSinduced by Z. We fix once and for all Y1, Y2⊂Xand w∈ASsuch that Yw 1=Y2. Suppose first that AX is of type Bn; observe that ASmust be of type F4or Bm(m >n) and that the first possibility might occur only if n ⩽3. We give a detailed proof assuming that ASis of type Bm; the case ASof type F4can be dealt with in a similar fashion and is left as an exercise for the reader. Given any subset Z⊂S, we denote by Zthe set of vertices of the connected component of ΓZcontaining s1and we set Z=∅if s1/∈Z; we also denote Z =Z\Z. Notice that the conjugation by winduces an isomorphism between the Coxeter graphs ΓY1 and ΓY2. Then Y 1=Y 2with yw=yfor all y∈Y 1(due to the defining relations of AS) and the graphs ΓY 1and ΓY 2have to be isomorphic. If Y 1is empty, we can replace w by the trivial element of AXand we are done. Otherwise observe that Y 1and Y 2are subsets of {s2, ..., sn}, which generates a braid group on nstrands. By applying Proposition 11, we can find w∈A{s2,...,sn}⊂AXperforming the same conjugation as won Y1 (if Y 1=∅, we can choose w∈A{s3,...,sn}commuting with Y 1). Suppose now that AXis of type Dn(nodd) and that ASis of type Dm(m>n). Recall (Table 1) that conjugation by ΔXleaves invariant s3, ..., snand permutes s1 and s2. If each of the chosen subsets Y1and Y2contains at most one of s1, s2, up to replacing one of Yi(i =1, 2) by YΔX i, we may assume that both Y1, Y2are subsets of {s1, s3, ..., sn}; the latter set defines a braid group of type An−1and Proposition 11 allows us to conclude. Suppose that Y1contains both s1and s2. Then Y2has to contain also both s1and s2. To see this, observe that the only ribbon adjacent to {s1, s2}is s3s1s2s3which conjugates s1to s2and s2to s1and apply Lemma 4: it follows that s1and s2can be simultaneously conjugated in ASto letters in Sonly if they are fixed or permuted with each other. As for type B, denoting by Y i(i =1, 2) the set of vertices of the (union of the) connected component(s) of ΓYicontaining s1and s2, we obtain Y 1=Y 2. We also M. Calvez et al. / Journal of Algebra 556 (2020) 621–633 629 see that the Y i=Yi\Y idefine isomorphic subgroups of the braid group A{s4,...,sn}, and we conclude as in type Bcase using Proposition 11 again.  4. Proof of Theorem 1 4.1. Irreducible case Let ASbe an irreducible Artin-Tits group of spherical type and let ΓSbe its defining Coxeter graph. Vertices of ΓSare numbered s1, ...s #S, according to Fig. 1. Let ∅ = XSsuch that AXis irreducible. First, we observe that, as AXis a proper subgroup of AS, it cannot be of type E8, F4, H4or I2m, m >5. Suppose that the pair (AX, AS)does not satisfy any of the conditions (a) to (d) of Theorem 1(1). The group AXcannot be either of type E7, D2kor H3; otherwise (AX, AS) would satisfy either (b), (c) or (d). Finally, AXcan be of type D5(D7, respectively) only if ASis of type Dn, n ⩾6, (n ⩾8, respectively); otherwise (AX, AS)would satisfy (a). Then Proposition 6, Proposition 11 and Proposition 12 show that AXis conjugacy stable in AS, as desired. Therefore, to prove the first part of Theorem 1, one has to prove that AXis not conjugacy stable in ASwhenever (AX, AS)satisfies one of the conditions (a) to (d). Lemma 4will be the main tool to provide counterexamples. In each case (a) to (d), we shall exhibit two elements of AXwhich are conjugate in ASbut not in AX. Let ΓXbe the defining Coxeter graph of AXand number the elements of X x1, ..., x#X, according to Fig. 1. Notice that there might be different ways to embed ΓX as an induced subgraph of ΓS; following our notation, this is to say that a given ximay be equal to distinct sj’s, depending on the chosen embedding of ΓXin ΓS. (a) Suppose that AXis of type D5and ASis of type E6, E7or E8. There are 4 different embeddings ι :Γ X→ΓS, namely: ι1:x1=s2,x 2=s3,x 3=s4,x 4=s5,x 5=s6, ι2:x1=s3,x 2=s2,x 3=s4,x 4=s5,x 5=s6, ι3:x1=s5,x 2=s2,x 3=s4,x 4=s3,x 5=s1, ι4:x1=s2x2=s5,x 3=s4,x 4=s3,x 5=s1. However to pass from one to another it suffices to pre- or post-compose by graph automorphisms which are induced by conjugation by ΔXor Δ{s1,s2,s3,s4,s5,s6}respectively. Therefore it is enough to give a pair of non-conjugate elements of AX whose images under ι1are conjugate elements of AS. Take g=ι1(x1x3x2) =s2s4s3and h =ι1(x4x3x2) =s5s4s3. The following product of ribbons (each arrow indicates the conjugation by its label) conjugates gto hin AS: