A Note on Commuting Involution Graphs in Affine Coxeter Groups

arXiv:1809.04834v1 [math.GR] 13 Sep 2018 A Note on Commuting Involution Graphs in Affine Coxeter Groups By Sarah Hart and Amal Sbeiti Clarke Birkbeck Mathematical Sciences Preprint Series Preprint Number 43 www.bbk.ac.uk/ems/research/pure/preprints A Note on Commuting Involution Graphs in Affine Coxeter Groups Sarah Hart and Amal Sbeiti Clarke 1 Introduction Let G be a group and X a set of involutions of G. The commuting involution graph C(G, X) is the graph whose vertex set is X, with vertices x, y joined by an edge whenever x and y commute. These graphs have been studied for a wide variety of groups, usually with X being either a conjugacy class or the set of all involutions. Perhaps the most well-known example of their use was in the work of Fischer on 3-transposition groups [5]. In a series of papers, Bates et al looked at connectedness and diameters of commuting involution graphs in the case where X is a conjugacy class of involutions and G is the symmetric group [1], a finite Coxeter group [2], a linear group [3], or a sporadic simple group [4]. In particular, for finite Coxeter groups they gave necessary and sufficient conditions under which the commuting involution graphs are connected, along with bounds for the diameters in the connected cases. For the symmetric group, where C(G, X) is connected the diameter is at most 4. For finite Coxeter groups this bound increases to 5. In [9], affine Coxeter groups of type Ãn were considered. Here, the diameter of any connected commuting involution graph is at most 6. More recently, Hart and Sbeiti Clarke considered the remaining classical Weyl groups (see [7], [6] and [11]). They showed that if G is a classical affine Weyl group, then if C(G, X) is connected, its diameter exceeds the rank of G by at most 1. The obvious next question is: what happens in the exceptional affine groups? The purpose of this short note is to establish some general results for commuting involution graphs in affine Coxeter groups, and to deal with types F̃4 and G̃2 . Types Ẽ6 Ẽ7 and Ẽ8 are more substantial and these will be addressed in a forthcoming paper. Section 2 contains preliminaries and general results. Section 3 deals with F̃4 and G̃2 . 2 Preliminaries and General Results Let W be a finite Weyl group with root system Φ in a Euclidean vector space V ∼ = Rn , and 2α , for α ∈ Φ. The affine Weyl group W̃ is the semidirect Φ∨ the set of coroots α∨ = hα,αi 1 product of W with translation group Z of the coroot lattice L(Φ∨ ) of W . We often express roots and coroots in terms of the standard basis {e1 , . . . , en }, and we will, by a slight abuse of notation, write elements of Z simply as vectors – in other words we will identify Z with L(Φ∨ ). For any w ∈ W̃ , w is written in the form (a, u) where a ∈ W and u ∈ Z. See, for example, [8, Chapter 4] for more detail. For a, b ∈ W and u, v ∈ Z we have (a, u)(b, v) = (ab, ub + v). We have (a, u)−1 = (a−1 , −ua ). In W̃ , the element (a, u) is conjugate to (b, v) via some (g, w) if and only if −1 (b, v) = (a, u)(g,w) = (g −1 ag, ug + w − wg −1 ag ). The reflections of W̃ are the affine reflections sα,k (α ∈ Φ, k ∈ Z). Recall that, for v in V , hα, vi α = v − hα, viα∨ hα, αi (hα, vi − k) sα,k (v) = v − 2 α = sα (v) + kα∨ . hα, αi sα (v) = v − 2 If R is a set of simple reflections for W and α̃ is the highest root (that is, the root with the highest coefficient sum when expressed as a linear combination of simple roots), then it can be shown that R ∪ {sα̃,1 } is a set of simple reflections for W̃ . Finally, we write Diam C(G, X) for the diameter of C(G, X) when C(G, X) is a connected graph, in other words the maximum distance d(x, y) between any x, y ∈ X in the graph. We begin with an observation about connectedness. For an element x = (a, u) in a conjugacy class X of W̃ , we define x̂ = a. Then let X̂ be the conjugacy class of x̂ in W . Clearly if x, y ∈ X, then x̂, ŷ ∈ X̂. Lemma 2.1. Suppose x, y ∈ X. If d(x̂, ŷ) = k, then d(x, y) ≥ k. If C(W, X̂) is disconnected, then C(W̃ , X) is disconnected. Proof. The result follows immediately from the observation that if x commutes with y in W̃ , then x̂ commutes with ŷ in W . Definition 2.2. Let W be an arbitrary Coxeter group, with R the set of simple reflections. Two subsets I and J of R are W -equivalent if there exists w ∈ W such that I w = J. In the next result, we use the notation wI for the longest element of a finite standard parabolic subgroup WI . 2 Theorem 2.3 (Richardson [10]). Let W be an arbitrary Coxeter group, with R the set of simple reflections. Let g ∈ W be an involution. Then there exists I ⊆ R such that wI is central in WI , and g is conjugate to wI . In addition, for I, J ⊆ R, wI is conjugate to wJ if and only if I and J are W -equivalent. For the rest of this paper, W will always denote a finite Weyl group with root system Φ, with W̃ its corresponding affine Weyl group. We will use the convention that a Weyl group of type Γ will be denoted W (Γ), where Γ is the associated Coxeter graph. The next two lemmas give conditions under which reflections w and w′ of W̃ commute. Lemma 2.4. Let α, β ∈ Φ. Then sα commutes with sβ if and only if hα, βi = 0 or α = ±β. Proof Consider v ∈ V and α, β ∈ Φ. A quick calculation shows that 2hv, αi α− hα, αi 2hv, αi α− sα sβ (v) = v − hα, αi sβ sα (v) = v − 2hv, βi β+ hβ, βi 2hv, βi β+ hβ, βi 4hv, αihα, βi β; hα, αihβ, βi 4hv, βihα, βi α. hβ, βihα, αi Hence sα sβ = sβ sα if and only if either hα, βi = 0 or hv, βiα = hv, αiβ for all v ∈ V . One of the properties of root systems is that for any root γ we have hγi ∩ Φ = {±γ}. Therefore sα sβ = sβ sα if and only if hα, βi = 0 or α = ±β. Lemma 2.5. For all positive roots α and β, and all integers k and l, the affine reflections sα,k and sβ,l commute if and only if either hα, βi = 0, or α = β and k = l. Proof. Let α and β be positive roots, with k and l integers. Then sα,k sβ,l (v) = sα,k (sβ (v) + lβ ∨ ) = sα sβ (v) + sα (lβ ∨ ) + kα∨ = sα sβ (v) + kα∨ + lβ ∨ − lhα, β ∨ iα∨ ; sβ,l sα,k (v) = sβ sα (v) + kα∨ + lβ ∨ − khα∨ , βiβ ∨ . Consequently sα,k sβ,l = sβ,l sα,k precisely when, for all v ∈ V , we have sα sβ (v) − lhα, β ∨ iα∨ = sβ sα (v) − khα∨ , βiβ ∨ . In particular, setting v = 0 we get lhα, β ∨ iα∨ = khα∨ , βiβ ∨ , which implies that either α = β or hα, βi = 0. If α = β, we get sα,k sα,l (v) = v + (l − k)α∨ , whereas sα,l sα,k (v) = v + (k − l)α∨ . Therefore, sα,k commutes with sα,l if and only if k = l. On the other hand, if α 6= β, then hα, βi = 0 and so sα sβ = sβ sα . Thus sα,k sβ,l = sβ,l sα,k . There is one case we can deal with that occurs in several groups. Lemma 2.6. Suppose W is a finite Weyl group that has a central involution w̃. Now suppose x is an involution in the corresponding affine Weyl group W̃ such that x = (w̃, u) for some u ∈ Z, and write X = xW̃ , the conjugacy class in W̃ of x. Then C(W̃ , X) is disconnected. 3 Proof. Since w̃ is central in W , every conjugate of x in W̃ has the form y = (w̃, v) for some u in Z. Moreover w̃ acts as −1 on the root system, and hence on Z. Now xy = (w̃, u)(w̃, v) = (w̃2 , uw̃ + v) = (1, v − u). Thus xy is translation through v − u. Now x commutes with y precisely when (xy)2 = 1, and hence x commutes with y if and only if u = v, meaning x does not commute with any other member of its conjugacy class. Therefore C(W̃ , X) is completely disconnected. Next we have a result which proves connectedness in certain circumstances. We write x ↔ y to mean xy = yx. Proposition 2.7. Let X be a conjugacy class of involutions in W̃ that contains (a, 0) for some involution a of W . Suppose that C(W, X̂) is connected with diameter d, and further that there is an integer k such that whenever (a, u) ∈ X, there is some x̂ ∈ X̂ such that d((a, u), (x̂, 0)) ≤ k in C(W̃ , X). Then C(W̃ , X) is connected with diameter at most d + k. Proof. Let (b, v) ∈ X. Then b is conjugate to a in W . That is, there is some g ∈ W −1 with b = ag , meaning that (b, v)g is of the form (a, u) for an appropriate u. By hypothesis then, there is a path (a, u) = x0 ↔ x1 · · · ↔ xm = (x̂, 0) of elements xi of X, with m ≤ k. But this implies that there is a path xg0 ↔ xg1 ↔ · · · ↔ xgm in the commuting involution graph. Writing y = x̂g , and noting that xg0 = (b, v), we obtain d((b, v), (y, 0)) ≤ k. Now C(W, X̂) is connected with diameter d. Thus d(y, a) ≤ d in C(W, X̂). Hence d((y, 0), (a, 0)) ≤ d in C(W̃ , X). Therefore d((b, v), (a, 0)) ≤ d + k and since this holds for all (b, v) in X, we deduce that C(W̃ , X) is connected with diameter at most d + k. Lemma 2.8. Suppose a ∈ WI for some I ⊆ R. If (a, u) is an involution, with u = P u α /PI. Moreover, let J be the r∈R r r , then a is an involution and ur = 0 whenever r ∈ set of reflections s of R that commute with all r in I. If v = s∈J vs αs , then vb = v for all b ∈ WI . Proof. We have (a, u)(a, u) = (a2 , ua +u). If (a, u) is an involution, then a is an involution and ua + u = 0. Now a is a product of elements of I. We have vr = v − hv, αr iαr∨ for all u ∈ Z and r ∈ R. Hence, inductively, ua = u − x for some x ∈ hαr : r ∈ Ii. So ua + u = 2u − x. For this to equal zero, clearly u ∈ hαr : r ∈ Ii. That is, ur = 0 whenever r∈ / I. For the second part, observe that for any r ∈ I and s ∈ J we have hαr , αs i = 0. Therefore vr = v for all r ∈ I. Hence vb = u for all b ∈ WI . Lemma 2.9. Let (a, u) be an involution in W̃ , with X = (a, u)W̃ . Suppose b commutes with a in W , where b ∈ aW . Then there is (b, v) ∈ X such that (a, u) ↔ (b, v). Proof. Since b is conjugate in W to a, there is g ∈ W with b = g −1 ag. Let w = 21 (u − ug ). Then (a, u)(g,w) = (g −1 ag, ug + w − wg 4 −1 ag ) = (b, ug + w − wb ). Set v = ug + w − wb . We claim that (a, u) ↔ (b, v). Certainly (b, v) ∈ X. Note that since (a, u) is an involution, ua = −u; also since g −1 ag = b we have gb = ag and so ugb = uag = −ug . Thus b v = ug + w − wb = ug + 12 (u − ug ) − 21 (ub − ug ) = ug + 12 u − 12 ug − 21 ub − 12 ug = 12 (u − ub ). Now (a, u)(b, v) = (ab, ub + v) = (ab, ub + 12 (u − ub )) = (ab, 21 (u + ub )) (b, v)(a, u) = (ba, va + u) = (ab, 12 ua − 21 uba + u) = (ab, 21 (u − uab )) = (ab, 21 (u + ub )). Thus (a, u) ↔ (b, v), as required. Proposition 2.10. Let X be a conjugacy class of involutions in W̃ containing (a, u). Suppose there is an integer k such that whenever (a, u′ ) ∈ X, we have d((a, u′ ), (a, u)) ≤ k, and also that C(W, X̂) is connected with diameter d. Then C(W̃ , X) is connected with diameter at most d + k. Proof. Let (b, v) ∈ X. By hypothesis C(W, X̂) is connected with diameter d, meaning d(a, b) ≤ d. Hence, by Lemma 2.9, there is a corresponding path (b, v) ↔ · · · ↔ (a, u′ ), for appropriate u′ , in C(W̃ , X) of length at most d. By hypothesis d((a, u′ ), (a, u)) ≤ k. Therefore, d((b, v), (a, u)) ≤ d + k. 3 Types F̃4 and G̃2 Let W be of type F4 , with associated root system Φ in R4 and simple roots {α1 , α2 , α3 , α4 }. The root system Φ consists of the 24 long roots ±ei ± ej (1 ≤ i < j ≤ 4) and 24 short roots: eight of the form ±ei and sixteen of the form 21 (±e1 ± e2 ± e3 ± e4 ). We may set α1 = 12 (e1 − e2 − e3 − e4 ), α2 = e4 , α3 = e3 − e4 and α4 = e2 − e3 . The highest root α̃ is then e1 + e2 . Then W = hsα1 , sα2 , sα3 , sα4 i and for the simple reflections of W̃ we can take ri = (sαi , 0) for i ∈ {1, 2, 3, 4} and r5 = (sα̃ , α̃∨ ). The Coxeter graph for W̃ is as shown in Figure 1. (The subgroup hr1 , r2 , r3 , r4 i is of course isomorphic to W and we may for convenience identify it with W on occasion.) t r1 t r2 t r3 t r4 t r5 Figure 1: Coxeter graph for type F̃4 Information about the commuting involution graphs of W was obtained in [2]. There are seven conjugacy classes of involutions in W . Let a = (sα2 sα3 )2 . Then C(W, X) is connected with diameter 2. Apart from this, and the graph consisting of just the central 5 Graph A1 A1 A21 A21 B2 B3 B3 A31 B2 × A1 F4 B4 B3 × A1 Representative I {r1 } {r3 } {r1 , r3 } {r3 , r5 } {r2 , r3 } {r1 , r2 , r3 } {r2 , r3 , r4 } {r1 , r3 , r5 } {r2 , r3 , r5 } {r1 , r2 , r3 , r4 } {r2 , r3 , r4 , r5 } {r1 , r2 , r3 , r5 } Underlying class in W A1 A1 A21 B2 B2 B3 B3 B3 B3 F4 F4 F4 Table 1: Conjugacy classes in F̃4 involution, all the other commuting involution graphs are disconnected. See [2] for more details. Now let us consider the affine group W̃ . By Theorem 2.3, every involution conjugacy class corresponds to a standard parabolic subgroup WI with a central involution wI . We can identify the possibilities by finding subgraphs of the Coxeter graph for F̃4 which correspond to Coxeter groups having nontrivial centres. Table 1 shows, for each subgraph giving rise to a distinct involution conjugacy class X, a representative I ⊆ {r1 , . . . , r5 } for which wI ∈ X, along with the name, in the third column, of the Coxeter graph for the underlying class X̂ in W . To determine the Coxeter graph corresponding to X̂, note that if I ⊆ W , then X̂ has the same graph as X. If x = (a, u) is a product of k reflections in W̃ , then a must be a product of k reflections in W , so, for example, the conjugacy class of W̃ corresponding to the B4 graph must have the class corresponding to F4 as its underlying class in W . The only instance where this does not immediately tell us the type of the underlying class is the case of A21 when I = {r3 , r5 }. Here, the underlying class might be type B2 or type A21 . Note that r3 = se3 −e4 and r5 = se1 +e2 ,1 . So the underlying class is the conjugacy class of se3 −e4 se1 +e2 in W . One can check that r1 r4 r3 r2 r3 r4 (e1 +e2 ) = e3 +e4 and r1 r4 r3 r2 r3 r4 (e3 − e4 ) = e3 − e4 . Hence (se3 −e4 se1 +e2 )r4 r3 r2 r3 r4 r1 = se3 −e4 se3 +e4 = se3 se4 = r3 r2 r3 r2 = (r3 r2 )2 . Consequently the underlying conjugacy class in this case is of type B2 . Theorem 3.1. Let W̃ be of type F̃4 , with graph shown in Figure 1. If X is the conjugacy class of (r2 r3 )2 or r3 r5 in F̃4 , then C(W̃ , X) is connected with diameter at most 4. Otherwise, C(W̃ , X) is disconnected. Proof. Let X be a conjugacy class in W̃ . If the underlying class in W is anything other than type B2 or F4 , then C(W̃ , X) is disconnected, by Lemma 2.1. If the underlying class is 6 type F4 then C(W̃ , X) is disconnected by Lemma 2.6. So we are reduced to the case where the underlying class in W is type B2 . There are two classes in W̃ where this happens, one containing (r2 r3 )2 = ((sα2 sα3 )2 , 0), and one containing r3 r5 = ((sα2 sα3 )2 , α̃∨ ). Let X be the conjugacy class of (r2 r3 )2 in W̃ ; its underlying class X̂ in W is of type B2 . Note that X̂ contains ((sα2 sα3 )2 = se3 se4 and thus also se1 se2 (for example via the conjugating element se1 −e3 se2 −e4 ). In Proposition 2.7, set a = se1 se2 and x̂ = se3 se4 . Suppose (a, u) ∈ X. Now (a, u)(x̂, 0) = (se1 se2 se3 se4 , v) for the appropriate v. This is clearly an involution because se1 se2 se3 se4 acts as −1 on Z. Thus (x̂, 0) commutes with (a bu). Therefore we can apply Proposition 2.7 with k = 1 and d = 2, to see that C(W̃ , X) is connected with diameter at most 3. Now let X be the conjugacy class of r3 r5 in W̃ ; again its underlying class in W is of type B2 . Let (a, u), (a, u′ ) ∈ X, where again a = se3 se4 . By Lemma 2.9 there is some v ∈ Z for which (a, u) ↔ (se1 se2 , v). But now (a, u′ )(se1 se2 , v) = (se1 se2 se3 se4 , v′ ) for some v′ , which is an involution. Thus in Proposition 2.10 we have k = 2 and d = 2, meaning that C(W̃ , X) is connected with diameter at most 4. The Coxeter graph of type G̃2 is as follows. r1 r2 6 ✉ ✉ G̃2 r3 ✉ The subgraphs corresponding to parabolic subgroups WI for which wI is central are of types A1 , A1 × A1 or G2 . It turns out that in all these cases, the commuting involution graphs are disconnected. Proposition 3.2. Let X be a conjugacy class in the affine Coxeter group W̃ of type G̃2 . Then C(W̃ , X) is disconnected. Proof. When W̃ is of type G̃2 , the underlying Weyl group W is dihedral of order 12. It has three conjugacy classes of involutions: two classes of reflections each with three elements, and one class consisting of the unique central involution w̃. The commuting involution graphs for the classes of reflections in W are completely disconnected. Now let x ∈ X. If x̂ is a reflection, then C(G, X) is disconnected by Lemma 2.1. Otherwise, x̂ = w̃, and so the disconnectedness of C(G, X) follows from Proposition 2.6. References [1] C. Bates, D. Bundy, P. Rowley and S. Perkins. Commuting involution graphs for symmetric groups, J. Algebra, 266(1) (2003), 133–153. [2] C. Bates, D. Bundy, P. Rowley and S. Perkins. Commuting Involution Graphs for Finite Coxeter Groups, J. Group Theory 6 (2003), 461–476. [3] C. Bates, D. Bundy, P. Rowley and S. Perkins. Commuting involution graphs in special linear groups. Comm. in Algebra, 32(11) (2004), 4179-4196. 7 [4] C. Bates, D. Bundy, P. Rowley and S. Hart. Commuting involution graphs for sporadic simple groups. J. Algebra, 316(2) (2007), 849-868. [5] B. Fischer. Finite groups generated by 3-transpositions, I. Invent. Math. 13 (1971), 232–246. [6] S. Hart and A. Sbeiti Clarke. Commuting Involution Graphs for C̃n , Comm. in Algebra 46 (9) (2018), 3965–3985. [7] S. Hart and A. Sbeiti Clarke. Commuting Involution Graphs in Classical Affine Groups, Birkbeck Mathematical Sciences Preprint Series No. 42 (2018). http://www.bbk.ac.uk/ems/research/pure/preprints [8] J.E. Humphreys. Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29 (1990). [9] S. Perkins. Commuting Involution Graphs in the affine Weyl group Ã, Arch. Math. 86 (2006), no. 1,16-25. [10] R. W. Richardson. Conjugacy Classes of Involutions in Coxeter Groups, Bull. Austral. Math. Soc. 26 (1982), 1–15. [11] A. Sbeiti Clarke. Affine Coxeter Groups, Conjugacy Classes and Commuting Involution Graphs, Ph.D. Thesis, Birkbeck (University of London), 2018. 8