Commuting Involution Graphs for Ãn

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1 Commuting Involution Graphs for Ãn Sarah Perkins Abstract In this article we consider the commuting graphs of involution conjugacy classes in the affine Weyl group Ãn. We show that where the graph is connected the diameter is at most 6. MSC(2): 2F55, 5C25, 2D6. Introduction Let G be a group and X a subset of G. The commuting graph on X, denoted C(G, X), has vertex set X and an edge joining x, y X whenever xy = yx. If in addition X is a set of involutions, then C(G, X) is called a commuting involution graph. Commuting graphs have been investigated by many authors. Sometimes they are tools used in the proof of a theorem, or they may be studied as a way of shedding light on the structures of certain groups (as in []). Commuting involution graphs for the case where X is a conjugacy class of involutions were studied by Fischer [4] in that case X was the class of 3-transpositions of a 3-transposition group. These groups include all finite simply laced Weyl groups, in particular the symmetric group. Commuting involution graphs for arbitrary involution conjugacy classes of symmetric groups were considered in [2]. The remaining finite Coxeter groups were dealt with in [3]. In this article we consider commuting involution graphs in the affine Coxeter group of type Ãn. As in [2] and [3], we will focus on the diameter of these graphs. We show that if X is a conjugacy class of involutions, then either the graph is disconnected or it has diameter at most 6. For the rest of this paper, let G n denote Ãn, for some n 2, writing G when n is not specified, and let X be a conjugacy class of involutions of G. We write Diam C(G, X) for the diameter of C(G, X) (when it is connected). Let Ĝ be the underlying Weyl group A n of G. It will be shown that every conjugacy class X of G corresponds to a certain conjugacy class ˆX of Ĝ. We may now state our main results (notation will be explained in Section 3). Theorem. Let G = G n = Ã n and a = (2)(34) (2m 2m) ˆX. Then C(G, X)is connected unless n = 2m + or m = and n {2, 4}. Theorem.2 Suppose C(G, X) is connected. If n > 2m or m is even, then Diam C(G, X) Diam C(Ĝ, ˆX) + 2. If n = 2m and m is odd, then Diam C(G, X) Diam C(Ĝ, ˆX) + 3. Using results about commuting involution graphs in A n (see Section 2) we can then deduce the following result. Corollary.3 Let G = G n = Ã n and a = (2)(34) (2m 2m). Suppose C(G, X) is connected. (i) If n 2m + 2 or n >, then Diam C(G, X) 5. (ii) If n = 2m + 2 and n = 6, 8 or then Diam C(G, X) 6. In Section 2 we will establish notation, describe the conjugacy classes of involutions in G and state results which we will require. Section 3 is devoted to proving Theorem.2. In Section 4 we give examples of commuting involution graphs which show that the bounds of Theorem.2 are strict. School of Economics, Mathematics and Statistics, Birkbeck College, Malet Street, London, WCE 7HX. s.perkins@bbk.ac.uk

2 Remark In the case of finite Weyl groups, given any conjugacy class X of a finite Weyl group W, it was shown in [3] that if C(G, X) is connected, then Diam C(G, X) 5. It is natural to ask whether there is a similar bound in the case of affine Weyl groups. The answer is no. Let W = B n, and let W I be a standard parabolic subgroup of G such that W I has type B n. Let w I be the central involution of W I, and set X = wi W. It can be shown that Diam C(G, X) = n. Thus the set of diameters of commuting involution graphs is unbounded. 2 The group G n = Ã n Let W be a finite Weyl group with root system Φ and let ˇΦ denote the set of coroots. (For full details, see for example [5].) The affine Weyl group W is the semidirect product of W with the translation group Z of the coroot lattice ZˇΦ of W. Elements of W are written as pairs (w, z), for w W, z Z. Multiplication is given by (σ, v)(τ, u) = (στ, v τ + u). We now fix W = A n. Then W = Sym(n), the symmetric group of degree n. W acts on R n = ε, ε 2,... ε n by permuting the subscripts of the basis vectors. The root system Φ of W is the set {±(ε i ε j ) : i < j n}, and in this case ˇΦ = Φ. Writing a translation by n i= ε i as (λ,..., λ n ), we see that Z = (,...,,,,...,,,,..., ) n = (u,..., u n ) : u i =. i= 2. Involutions in G n By the definition of group multiplication in G n, we see that the element (σ, v) of G is an involution precisely when (σ 2, v σ + v) = (, ). So σ is an involution of Sym(n) and for appropriate a i, b i, c i and m, σ = (a b ) (a m b m )(c 2m+ )(c 2m+2 ) (c n ). Setting v = (v,, v n ) we must have Hence we have the following lemma: v a + v b = = v am + v bm = 2v c2m+ = = 2v cn =. Lemma 2. Any involution in G n is of the form (σ, v), where σ = (a b ) (a m b m )(c 2m+ )(c 2m+2 ) (c n ), with v bi = v ai for i m and v ci = for 2m + i n. It will be convenient to use a more compact notation for involutions of G. Let g = ( m i= (α iβ i ), v) with α i, β i {,..., n} for i m. Then, by Lemma 2., v βi = v αi, and if j / {α,..., α m, β,..., β m }, then v j =. Thus v is determined from the set := v αi, i m. We may therefore write g = m (α i β i ). i= 2

3 2.2 Conjugacy classes of Involutions We now describe the conjugacy classes of involutions in G n. Conjugacy classes of involutions in Coxeter groups are well understood and in order to use the known results we must give another description of G n, this time in terms of its Coxeter graph. A Coxeter group W has a generating set R of involutions (known as the fundamental reflections), where the only relations are (rs) mrs = (r, s R), with m rr = and, for r s, m rs = m sr 2. This information is encoded in the Coxeter graph Γ = Γ(W ). The vertex set of Γ is R, where vertices r, s are joined by an edge labelled m rs whenever m rs > 2. By convention the label is omitted when m rs = 3. The Coxeter graphs of G 2 = Ã and G n = Ã n, n 3 are as follows: G 2 r r 2 G n (n 3) r n r r 2 r n We may define r n = (n), and for i n, r i = (i i + ) (using the notation defined in Section 2.2). It is not difficult to see that the appropriate relations hold. The symmetric group Sym(n) is a Coxeter group of type A n, with Coxeter graph Sym(n) = A n r r 2 r n We may set r i = (i i + ) for i n. Definition 2.2 Let W be an arbitrary Coxeter group, with I, J two subsets of R. We say that I, J are W -equivalent if there exists w W such that I w = J. Any subset I of R generates a Coxeter group in its own right, denoted W I. Such subgroups are called standard parabolic subgroups of W. If W I is finite then it has a unique longest element, denoted w I. Richardson [6] proved Theorem 2.3 Let W be an arbitrary Coxeter group, with R the set of fundamental reflections. Let g W be an involution. Then there exists I R such that w I is central in W I, and g is conjugate to w I. In addition, for I, J R, w I is conjugate to w J if and only if I and J are W -equivalent. It will be useful to narrow down the possible elements in the conjugacy class of involutions (a, u) in the case where a is an involution of Sym(n) with no fixed points. Lemma 2.4 Suppose n = 2m. Let a = m i= (α i β i ) and b = m i= (γ i δ i ). Suppose g = (a, u) and h = (b, v) are conjugate involutions of G n. Then m i= u α i m i= v γ i mod 2. Proof Let g = (a, u), and suppose h = (b, v) is conjugate to g in G n via (c, w). Reordering if necessary, assume that c(α i ) = γ i and c(β i ) = δ i for i m. We see that (b, v) = (a, u) (c,w) = (c ac, (w ) c ac + u c + w). Thus b = c ac and v = w w b + u c. Hence, for i m, v γi = w γi w b(γi ) + [u c ] γi. Since c(α i ) = γ i, it follows that [u c ] γi = u αi. Hence, recalling that n j= =, 3

4 m v γi = i= = m m (w γi w δi + u αi ) = (w γi + w δi 2w δi + u αi ) i= n m w j 2 w δi + i= m u αi j= i= i= i= m u αi mod 2. Therefore m i= u α i m i= v γ i mod 2, and the result holds. We use Theorem 2.3 to establish the next result. Proposition 2.5 Let g G be an involution. Then there is m Z + such that g is conjugate to exactly one of the following: If n = 2m and g = m mod 2. i= (2) (2m 2m); or (2)(34) (2m 2m) (and n = 2m). (α i β i ), then g is conjugate to (2) (2m 2m) if and only if m i= Proof By Theorem 2.3, g is conjugate to w I for some finite standard parabolic subgroup W I of W in which w I is central. Note that if g is also conjugate to w J for some J R then I = J, so that I only depends on g and not the particular choice of I. For any proper subset I R, we see that W I is isomorphic to a direct product of symmetric groups. The only symmetric group with non-trivial centre is Sym(2) = A. So for w I to be central, W I must be a direct product of symmetric groups of degree 2, and w I = r i r i2 r il for some l, where r ij r ik = r ik r ij for j < k l. This immediately implies that I n/2. Suppose that w I and w J are central in W I, W J respectively, and, in addition, that there exists r R\(I J). Set K = R\{r}. Then K is isomorphic to Sym(n). It is well known that conjugacy classes in the symmetric group are parameterised by cycle type, so that w I is conjugate to w J precisely when I = J. Therefore, in the case I < n/2, we may assume that I = {r, r 3,, r 2m } for some m < n/2, so that w I = (2) (2m 2m), with 2m < n. It only remains to consider the case I = n/2 (and then of course n must be even). We quickly see that there are only two possibilities for I such that w I is central in W I. Either I = {r, r 3,..., r n } and w I = g := (2) (2m 2m), or I = {r 2, r 4,..., r n } and w I = g 2 := (n) (23) (2m 2 2m ). By Lemma 2.4, g is not conjugate to g 2. Hence g is conjugate to exactly one of g and g 2. By Lemma 2.4, g 3 := (2) (34) (2m 2m) must be conjugate to g 2. Hence g = m i= (α i β i ) is conjugate to exactly one of g and g 3. Furthermore g is conjugate to g if and only if n i= mod 2. We have now proved Proposition 2.5. It can easily be seen that in the case n = 2m, the two conjugacy classes have isomorphic commuting involution graphs. Thus we may assume that g is conjugate to (2) (2m 2m). We end this section by stating some results from [2] concerning the diameters of commuting involution graphs in Sym(n). Let a = (2)(34) (2m 2m) Sym(n) and write Y = a Sym(n). Theorem 2.6 (Theorem. of [2]) C(Sym(n), Y ) is disconnected if and only if n = 2m + or n = 4 and m =. 4

5 Proposition 2.7 (Corollary 3.2 of [2]) If n = 2m, then C(Sym(n), Y ) is connected and Diam C(Sym(n), Y ) 2, with equality when n > 4. Theorem 2.8 (Theorem.2 of [2]) Suppose that C(Sym(n), Y ) is connected. Then one of the following holds: (i) Diam C(Sym(n), Y ) 3; or (ii) 2m + 2 = n {6, 8, } and Diam C(Sym(n), Y ) = 4. 3 Proof of Theorems. and.2 From now on, fix a = (2) (2m 2m), where 2m n, and set t = (a, ) = (2) (2m 2m) and X = t G. As we have observed, every commuting involution graph of G is isomorphic to C(G, X) for an appropriate choice of m. Write Ĝ = Sym(n) and ˆX = aĝ. Finally, if g = m i= (α i β i ) X, then set ĝ = m i= (α iβ i ) Ĝ. Clearly if g, h X, then ĝ, ĥ ˆX. We begin with the following lemma. Lemma 3. Suppose g, h X. If d(ĝ, ĥ) = k, then d(g, h) k. If C(Ĝ, ˆX) is disconnected, then C(G, X) is disconnected. Proof follows. Observe that if σ commutes with τ in G n then ˆσ commutes with ˆτ in Sym(n). The lemma Lemma 3.2 Let g = λ (αβ) λ 2 (γδ), g 2 = {,..., n} and integers, µ i. Then µ µ 2 (αγ)(βδ), g 3 = λ (αβ), g 4 = λ 2 (αβ) for distinct α, β, γ, δ in (a) g g 2 = g 2 g if and only if µ λ = µ 2 λ 2 ; (b) g 3 g 4 = g 4 g 3 if and only if λ = λ 2 ; (c) If h G is an involution such that ĥ(α) = α and ĥ(β) = β, then g 3h = hg 3 for all λ Z. Proof For part (a), we lose no generality by assuming, for ease of notation, that n = 4, g = λ (2) λ 2 (34) µ µ 2 and g 2 = (3)(24). That is, g = ((2)(34), (λ, λ, µ, µ )) and g 2 = ((3)(24), (λ 2, µ 2, λ 2, µ 2 )). Hence g g 2 = ((4)(23), (λ, λ, µ, µ ) (3)(24) + (λ 2, µ 2, λ 2, µ 2 )) = ((4)(23), (µ + λ 2, µ + µ 2, λ λ 2, λ µ 2 )). Now g and g 2 commute if and only if g g 2 is an involution. This occurs if and only if µ + λ 2 = ( λ µ 2 ) and µ + µ 2 = (λ λ 2 ). Rearranging gives µ λ = µ 2 λ 2, as required. For part (b), we may assume that n = 2, g 3 = ((2), (λ, λ )) and g 4 = ((2), (λ 2, λ 2 )). Then g 3 g 4 = (, ( λ + λ 2, λ λ 2 )). Hence g 3 g 4 = g 4 g 3 if and only if λ = λ 2. For part (c), we again assume that g 3 = ((2), (λ, λ )), and write h = (b, (v,..., v 2 )). Since b fixes and 2 and h is an involution, we must have v = v 2 =. Hence hg 3 = ((2)b, (λ, λ, v 3,..., v n ) = g 3 h. This completes the proof of Lemma 3.2. We may now dispose of the case n = 2. Proposition 3.3 Let G = G 2 = Ã. Then there are two conjugacy classes of involutions, representatives of which are (2) and (2). In either case C(G, X) is completely disconnected (the graph has no edges). 5

6 λ λ 2 A double transposition (α β )(α 2 β 2 ) for which λ + λ 2 is even is called an even pair. Otherwise it is an odd pair. Proposition 3.4 Suppose n > 2 and that C(Ĝ, ˆX) is connected. Let g X. If n > 2m or m is even, then there exists h = (c, ) X such that d(g, h) 2. Proof Let g X. We will find it useful to split g into various components. Since C(Ĝ, ˆX) is connected, by Theorem 2.6 either n = 2m or there are at least two fixed points. Recalling that either n > 2m or m is even, it is easily seen that we may write g in the following form: g = P P 2 P k Q where P,..., P k are even pairs and Q is either the identity (if n = 2m and m is even), a single transposition along with at least two fixed points, or an odd pair along with at least two fixed points. µ i ν i Let P i = (α i β i )(γ i δ i ) with µ i + ν i = 2 even. Now set P that each of P i, P i i = (α i δ i )(γ i β i ) and P and P i is an even pair. It is clear by Lemma 3.2(a) that P i P i = P 3.2(a) also implies that P i P i = P i P i, because we may rewrite P i = i = µ i (α i β i ) ν i (δ i γ i ) and P i = (α i γ i )(δ i β i ). Note i P i. But Lemma (α i δ i ) (β i γ i ). If Q is the identity, then let Q = Q = Q. If Q is a single transposition along with at least two fixed points, then we may write Q = (αβ) λ (ε ) (ε l ) for some l 2. Let Q = Q = (ε ε 2 )(α) (β) (ε 3 ) (ε l ). µ If Q is an odd pair, along with at least two fixed points, then we may write Q = (αβ)(γδ) ν (ε ) (ε l ) for some l 2. Let Q = (ε ε 2 )(γδ) ν (α) (β) (ε 3 ) (ε l ) and let Q = (αβ) (ε ε 2 )(γ) (δ) (ε 3 ) (ε l ). Then set g = P P k Q and h = P P k Q. Let c = ĥ. Then by choice of h, h = (c, ). If n = 2m, note that g and h consist entirely of even pairs, so g, h X. If n 2m, then g and h are obviously in X. By construction, and Lemma 3.2, g commutes with g and g commutes with h. Therefore d(g, h) 2. This completes the proof of the proposition. Proposition 3.5 Suppose that n = 2m with m > odd. Let g = (b, v) X. Then there exists h = (c, ) X such that d(g, h) 3. Proof µ We may write g = P Q where P is a product of k even pairs and Q is an even triple with µ 2 µ 3 Q = (α β )(α 2 β 2 )(α 3 β 3 ) and µ + µ 2 + µ 3 = 2λ for some λ Z. Set ρ = µ λ. µ ρ+µ 2 ρ+µ 3 Now define Q = (α β )(α 2 α 3 )(β 2 β 3 ), Q 2 = (α α 2 )(β α 3 )(β 2 β 3 ) and Q 3 = (α α 2 )(β β 3 )(α 3 β 2 ). By repeated use of Lemma 3.2(a), we see that QQ = Q Q, Q Q 2 = Q 2 Q and Q 2 Q 3 = Q 3 Q 2. Note also that Q, Q 2 and Q 3 are all even triples. By Proposition 3.4, there exist P, P, both products of k even pairs, such that P = 2k i= (γ i δ i ) for some γ i, δ i and P P = P P, P P = P P. In addition, we may assume that Fix( ˆP ) = Fix( ˆP ) = Fix( P ˆ ). Define g = P Q, g 2 = P Q 2 and h = P Q 3. Then by construction g commutes with g and g 2, and g 2 commutes with h. So d(g, h) 3. Furthermore, by construction g, g 2 and h are all elements of X, and h = (ĥ, ). We have now proved Proposition 3.5. Lemma 3.6 Suppose g = (b, ), g 2 = (b 2, ) X. If C(Ĝ, ˆX) is connected, then d(g, g 2 ) = d(b, b 2 ). We are now able to prove Theorems. and.2. µ 2 λ ρ+µ 3 6

7 Proof of Theorem. The case n = 2, m = is Proposition 3.3. If n = 2m + or n = 4, m =, then C(G, X) is disconnected by Lemma 3. and Theorem 2.6. If n > 2 and C(Ĝ, ˆX) is connected, then the fact that C(G, X) is connected is an easy consequence of Propositions 3.4 and 3.5, and Lemma 3.6. Proof of Theorem.2 By Proposition 3.4, if n > 2m or m is even, and C(G, X) is connected, then there exists h = (c, ) X such that d(g, h) 2. If n = 2m and m is odd, then by Proposition 3.5 there exists h = (c, ) X such that d(g, h) 3. By Lemma 3.6, d(h, t) Diam C(Ĝ, ˆX). Thus d(g, t) Diam C(Ĝ, ˆX) + 2 if n > 2m or m is even, and d(g, t) Diam C(Ĝ, ˆX) + 3 otherwise. Theorem.2 follows immediately. Corollary.3 now follows from Theorem.2 in conjunction with Proposition 2.7 and Theorem Two Examples In this section we give C(G, X) for two examples: n = 4, m = 2 and n = 6, m = 3. These graphs, of diameters 3 and 5 respectively, illustrate the fact that the bounds in Theorem.2 are tight, because the respective diameters of C(Sym(4), (2)(34) Sym(4) ) and C(Sym(6), (2)(34) Sym(6) ) are and 2. Figure shows C(G, X) for n = 4, m = 2. The variable(s) above a transposition can be taken to be λ any integers. So for example (3)(24) λ µ µ commutes with (2)(34) for any integers λ, µ. (2)(34) λ (3)(24) λ λ (4)(23) λ λ+µ (4) λ µ (23) µ (2)(34) µ µ (2)(34) µ λ+µ (3) λ µ (24) ρ+µ (2)(34) µ ρ ρ+µ (2)(34) ρ µ Figure : n = 4, m = 2 Figure 2 shows the collapsed adjacency graph in the case n = 6, m = 3. If g, h G are in the same orbit of the centralizer C G (t) of t in G, then clearly d(g, t) = d(h, t). The vertices of the graph in Figure 2 are the C G (t)-orbits of C(G, X). We give one representative for each C G (t)-orbit. 7

8 (2)(34) (3)(24) (4)(23) 2 (3)(25) 3 2 (46) (2)(34) 3 (4)(25) 3 (36) 6 (2)(35) 3 (46) 6 (2)(34) 3 6 (2)(34) (2)(35) 3 (46) 2 (2)(34) 3 2 Figure 2: n = 6, m = 3 References [] M. Aschbacher. Sporadic Groups, Cambridge Tracts in Mathematics, 4, Cambridge University Press, Cambridge (994). [2] C. Bates, D. Bundy, S. Perkins and P. Rowley. Commuting Involution Graphs for Symmetric Groups, J. Algebra 266 (23), [3] C. Bates, D. Bundy, S. Perkins and P. Rowley. Commuting Involution Graphs for Finite Coxeter Groups, J. Group Theory 6 (23), [4] B. Fischer, Finite groups generated by 3-transpositions, I. Invent. Math. 3 (97), [5] J.E. Humphreys. Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29 (99). [6] R.W. Richardson. Conjugacy Classes of Involutions in Coxeter Groups, Bull. Austral. Math. Soc. 26 (982), 5. 8