Braids with Trivial Simple Centralizer

Algebra Colloquium 2? : ? (201?) ??–??

Algebra Colloquium c 2013 AMSS CAS ° & SUZHOU UNIV

Braids with Trivial Simple Centralizer U. Ali†

F. Azam

I. Javaid

Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University, Multan, Pakistan E-mail: [email protected] ([email protected]) [email protected] [email protected]

A. Haider Department of Mathematics, Jazan University, Jazan, Saudi Arabia E-mail: [email protected] Received 5 December 2011 Revised 19 August 2012 Communicated by L.A. Bokut Abstract. In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤ n − 2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnected. We also prove that the graph has one major component. 2010 Mathematics Subject Classification: 11B39, 05A15, 05A05 Keywords: simple braids, simple centralizer, commuting graphs

1 Introduction A positive n-braid is a word in the set of generators {x1 , x2 , . . . , xn−1 } : 1 xi

2

i ...

i+1

HH© . . . © H

n−1

n

The set of positive n-braids or the braid monoid MB n is embedded in the braid group Bn . Artin [3] showed that MBn admits the presentation: D E x xx = xi xi+1 xi MB n = x1 , x2 , . . . , xn−1 : i+1 i i+1 . (1) xi xj = xj xi for |i − j| ≥ 2 A simple braid contains a letter xi at most once (see [4]). The set of simple braids SBn can be connected with Fibonacci numbers 1, 1, 2, 3, 5, . . ., and it is shown in [5] that the number of simple braids in SBn is the Fibonacci number F2n−1 . For example, there are 13 elements in SB4 : {e, x1 , x2 , x3 , x1 x2 , x1 x3 , x2 x1 , x2 x3 , x3 x2 , x1 x2 x3 , x1 x3 x2 , x2 x1 x3 , x3 x2 x1 }. †

Supported by HEC, Pakistan (Ref no: PM-IPFP/HRD/HEC/2010/1508).

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U. Ali, F. Azam, I. Javaid, A. Haider

Three kinds of divisors of β ∈ MB n are defined: divisors of β (γ|β), Div(β) = {γ ∈ MB n : ∃ δ, ε ∈ MB n , β = δγε}; left divisors of β (γ|L β), DivL (β) = {γ ∈ MB n : ∃ ε ∈ MB n , β = γε}; right divisors of β (γ|R β), DivR (β) = {γ ∈ MB n : ∃ δ ∈ MB n , β = δγ}. Clearly, DivL (β) ∪ DivR (β) ⊆ Div(β). Let ∆n = x1 (x2 x1 ) · · · (xn−1 xn−2 · · · x2 x1 ) be the Garside braid (see [4] for more details). The set SBn is a proper subset of Div(∆n ). The braid x1 x3 x2 x4 is simple while the braid x1 x3 x2 x3 is a non-simple divisor of ∆n for n ≥ 5. 1

2

HH© © H

1

3

4

H © HH © HH© © H

2

3

1

2

HH© © H

H © ©HH

4

β = x1 x3 x2 x4

5

5

1

3

4

5

H © ©HH H © ©HHH © ©HH

2

3

4

5

α = x1 x3 x2 x3

There is a canonical group homomorphism onto the symmetric group π : Bn → Σn ; ³ ´ 1 2 3 4 5 for example, π(α) = 4 1 3 2 5 . The restriction of π to Div(∆n ) is a bijection (see [14, Chapter 9]). The symmetric group Σn admits a presentation in the Coxeter generators: D E s2 = 1, si+1 si si+1 = si si+1 si Σn = s1 , s2 , . . . , sn−1 : i . si sj = sj si for |i − j| ≥ 2

(2)

The homomorphism π can be defined by π(xi ) = si . The image π(SBn ) = SΣn is called the set of simple permutations in Σn (see [6]). The braid monoid MB n satisfies left and right cancellation laws (see [15]). The monoid MBn is embedded in MBn+1 , and consequently, α, β ∈ MB n commute in MB n+1 if and only if α and β commute in MB n . Definition 1.1. [1] The simple centralizer of β ∈ SBn is the set Cn (β) = {γ ∈ SBn : βγ = γβ}, i.e, the intersection of centralizer of β in MB n with SBn . We say that a simple braid β ∈ SBn has trivial simple centralizer if Cn (β) = {e, β}. The cardinality of Cn (β) is denoted by cn (β). Theorem 1.2. [1] For any xi ∈ SBn , Cn (xi ) = {β ∈ SBn : xj - β for |j − i| = 1}. Our main result in this paper completely describes the structure of some special simple braids. For n ≥ 4, we have: Theorem 1.3. If β ∈ SBn and xi |β for all 2 ≤ i ≤ n − 2, then Cn (β) = {e, β}. Definition 1.4. [18, page 248] A graph is said to be planar if it can be embedded in a plane. Otherwise, it is non-planar. Definition 1.5. [7, 10, 16] A commuting graph Γ(H) associated to a group G and a finite subset H of G is a graph whose vertices are the elements of H\{e} and there is an edge between g and h if and only if g 6= h and gh = hg.

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Braids with Trivial Simple Centralizer

Some properties of the commuting graphs Γ(SBn ), Γ(SΣn ) and Γ(Σn ) were discussed in [1], for example: Proposition 1.6. [1] a) Γ(SBn ) is planar if and only if n ≤ 5. b) Γ(SΣn ) and Γ(Σn ) are planar if and only if n ≤ 4. As a direct consequence of Theorem 1.3, the graph Γ(SBn ) is disconnected. Theorem 1.3 is not true for simple permutations SΣn in Σn , because for every P1 = si1 si2 · · · sik , we have P2 = sik sik−1 · · · si1 in the simple centralizer of P1 , and consequently, Γ(SBn ) is a proper subgraph of Γ(SΣn ). Definition 1.7. A component of a graph other than an isolated vertex is called a major component. Theorem 1.3 also facilitates in proving the following: Proposition 1.8. The graph Γ(SBn ) has a unique major component for n ≥ 4. The next figure shows the major component of Γ(SB5 ): x1x2

x 4 x3

x2x1

x3x4 x3

x2 x2 x4

x2x1x4 x1x2x4

x4

x1 x1x4

x1x3

x1x3 x4 x1x4 x3

2 Gr¨ obner-Shirshov Bases The references [2, 8, 9, 11, 12, 13, 17] contain the following notions under different names: Gröbner-Shirshov bases, complete presentations, Non-commutative Gröbner bases, rewriting systems, presentations with solvable ambiguities and so on. Let F [Y ] be a free monoid generated by Y = {y1 , y2 , . . . , yn }. The total order on the set of generators given by y1 < y2 < · · · < yn is extended to length-lexicographic order < . Let M be a monoid, constructed by defining some relations in F [Y ]. A defining relation R in M is written in the form ai = bi where ai is a monomial greater than bi . In the monoid M, a word containing the L.H.S. of a relation is called reducible and a word which does not contain the L.H.S. of a relation is called irreducible. The diamond lemma (see [9] or [13]) says that if all the ambiguities are solvable (for any overlap ai = γi δ, aj = δ²j , δ 6= 1, reductions of (γi δ)²j → bi ²j → · · · and γi (δ²j ) → γi bj → · · · give the same result), then the set of irreducible words is in bijection with the monoid with the presentation hy1 , y2 , . . . , yn : ai = bi , i ∈ Ii (I could be an infinite set). The above presentation is called a Gröbner-Shirshov basis (complete presentation) and the irreducible words are called canonical forms. In this way, the word problem (deciding whether two words represent the same element or not) in the monoid M is solved by computing the canonical forms. The canonical form of a word W is denoted by cf (W ).

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For a word α(j, i) = α(xj , xj+1 , . . . , xi ) in xj , xj+1 , . . . , xi , we use the notation Σα(j, i) = α(xj+1 , xj+2 , . . . , xi+1 ). Theorem 2.1. [8, 11] A Gröbner-Shirshov basis (complete presentation) of MBn consists of the following relations: (i) xi+1 xi α(1, i − 1)β(j, i)λ(j, i + 1) = xi xi+1 xi α(1, i − 1)λ(j, i)Σβ(j, i), (ii) xs xk = xk xs (s − k ≥ 2), where 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ i + 1. 3 Proofs Since deciding whether two words α and β commute is equivalent to decide whether αβ and βα represent the same element of a monoid. We use Theorem 2.1 to compute the canonical forms cf (βα) and cf (αβ) for β, α ∈ SB4 , where β is a simple braid mentioned in Theorem 1.3 and α is any simple braid (other than the identity braid). For example, the following table contains canonical forms cf (βα) and cf (αβ) for β ∈ {x2 x3 , x3 x2 }. β cf (βα) cf (αβ) x2 x3 x2 x1 x3 x1 x2 x3 x2 x3 x2 x3 x2 x22 x3 x2 x3 x2 x23 x2 x3 x2 x2 x3 x2 x1 x3 x2 x1 x22 x3 2 x2 x3 x2 x1 x3 x1 x2 x3 x2 x2 x3 x2 x3 x2 x1 x1 x2 x1 x3 x2 x3 x2 x23 x2 x3 x22 x3 x2 x3 x1 x2 x1 x3 x2 x1 x22 x3 x2 x2 x3 x2 x1 x23 x2 x1 x3 x22 x3 x2 x3 x22 x3 x2 x1 x1 x2 x1 x3 x2 x2 x3 x2 x23 x2 x1 x1 x2 x3 x2 x1

α x1 x2 x3 x1 x2 x1 x3 x2 x1 x3 x2 x1 x2 x3 x1 x3 x2 x2 x1 x3 x3 x2 x1

β cf (βα) cf (αβ) x3 x2 x3 x2 x1 x1 x3 x2 x3 x2 x3 x22 x2 x3 x2 x3 x2 x2 x3 x2 x23 x2 x3 x2 x1 x3 x2 x1 x1 x2 x3 x2 x3 x2 x2 x3 x2 x1 x1 x23 x2 2 x3 x2 x3 x2 x1 x2 x1 x3 x2 x3 x2 x3 x22 x3 x2 x23 x2 x3 x2 x1 x2 x3 x2 x1 x1 x2 x23 x2 x3 x2 x2 x1 x3 x2 x1 x1 x2 x3 x22 x3 x2 x3 x22 x1 x3 x2 x1 x23 x2 2 x3 x2 x2 x3 x2 x1 x2 x1 x3 x2 x1

α x1 x2 x3 x1 x2 x1 x3 x2 x1 x2 x3 x1 x2 x3 x1 x3 x2 x2 x1 x3 x3 x2 x1

Similarly canonical forms cf (βα) and cf (αβ) can be computed for β = x2 , x1 x2 , x2 x1 , and one can see that βα 6= αβ. This accomplishes the following lemma. Lemma 3.1. If x2 |β in SB4 then C4 (β) = {e, β}. The following known lemmas will be used in our proofs. Lemma 3.2. [15, Theorems H and K] a) In MB n , given xi β1 = xj β2 , if |i − j| ≥ some γ ∈ MB n ; if |i − j| = 1, then β1 γ ∈ MB n . b) In MB n , given β1 xi = β2 xj , if |i − j| ≥ some γ ∈ MB n ; if |i − j| = 1, then β1 γ ∈ MB n .

2, then β1 = xj γ and β2 = xi γ for = xj xi γ and β2 = xi xj γ for some 2, then β1 = γxj and β2 = γxi for = γxi xj and β2 = γxj xi for some

Lemma 3.3. [1] Let β, γ ∈ MB n . a) If xn−1 |L βγ and xn−1 - β, then xn−1 |L γ. b) If xn−1 |R γβ and xn−1 - β, then xn−1 |R γ. Lemma 3.4. [1] If β ∈ SBn and xn−1 |β, then either xn−1 |R β or xn−1 |L β. Lemma 3.5. [1] If β ∈ SBn−1 and α ∈ SBn such that xn−2 |β and xn−1 |α, then βα 6= αβ.

Braids with Trivial Simple Centralizer

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Lemma 3.6. [1] If β(γ1 γ2 ) = (γ1 γ2 )β and βγ1 = γ1 β or βγ2 = γ2 β, then βγ2 = γ2 β or βγ1 = γ1 β, respectively. Lemma 3.7. [1] If α ∈ SBn−1 and xn−2 |α, then xn−1 - β for any β ∈ Cn+m (α), where m ≥ 1. Proof of Theorem 1.3. We use induction on n. The case for n = 4 is clear by Lemma 3.1. Let n ≥ 5 and suppose that the theorem is true for all β ∈ SBn−1 with the property that xi |β for all 2 ≤ i ≤ n − 3. For β ∈ SBn with the property that xi |β for all 2 ≤ i ≤ n − 2 and α ∈ Cn (β), we show that α ∈ {e, β}. If xn−1 - β, then β ∈ SBn−1 with xn−2 |β. We must have xn−1 - α by Lemma 3.5 and hence α ∈ {e, β} by induction. Now if xn−1 |β, we have to deal with two cases: a) xn−1 - α and b) xn−1 |α. For a), we also have xn−2 - α by Lemma 3.7. Hence, αxn−1 = xn−1 α by Theorem 1.2. We have αβ1 = β1 α by Lemmas 3.4 and 3.6, where either β = xn−1 β1 or β = β1 xn−1 , which shows that α ∈ {e, β1 } by induction. If α = β1 , then xn−1 β1 = β1 xn−1 which is a contradiction by Theorem 1.2, so we have α = e. For b), let α ∈ / {e, β}. Without loss of generality we can suppose by Lemma 3.4 that β = β1 xn−1 for β1 ∈ SBn−1 such that xi |β1 for all 2 ≤ i ≤ n − 2. Using Lemma 3.4, we have to deal with ten possibilities, where α2 , β2 ∈ SBn−2 such that xi |β2 for all 2 ≤ i ≤ n − 3: 1) α = xn−1 xn−2 α2 and β = β2 xn−2 xn−1 ; 2) α = xn−1 xn−2 α2 and β = xn−2 β2 xn−1 ; 3) α = xn−2 α2 xn−1 and β = β2 xn−2 xn−1 ; 4) α = xn−2 α2 xn−1 and β = xn−2 β2 xn−1 ; 5) α = xn−1 α2 xn−2 and β = β2 xn−2 xn−1 ; 6) α = xn−1 α2 xn−2 and β = xn−2 β2 xn−1 ; 7) α = α2 xn−2 xn−1 and β = β2 xn−2 xn−1 ; 8) α = α2 xn−2 xn−1 and β = xn−2 β2 xn−1 ; 9) α = α2 xn−1 and β = β2 xn−2 xn−1 ; 10) α = α2 xn−1 and β = xn−2 β2 xn−1 . We prove 1) and 9). The other cases can be proved with almost similar arguments. For 1), xn−1 xn−2 α2 β2 xn−2 xn−1 = β2 xn−2 x2n−1 xn−2 α2 and α2 xn−1 = xn−1 α2 imply xn−1 xn−2 α2 β2 xn−2 x2n−1 = β2 xn−2 xn−1 xn−1 xn−2 xn−1 α2 . By the presentation (1), xn−1 xn−2 α2 β2 xn−2 x2n−1 = xn−1 β2 xn−2 x2n−1 xn−2 α2 . By β2 xn−1 = xn−1 β2 and cancellation, xn−2 α2 β2 xn−2 x2n−1 = β2 xn−2 x2n−1 xn−2 α2 . By comparing the above words and applying cancellation, we have xn−2 α2 β2 xn−2 xn−1 = xn−1 xn−2 α2 β2 xn−2 . By Lemma 3.2, we get α2 β2 xn−2 = xn−1 γ for some γ ∈ MB n which is a contradiction since xn−1 - α2 β2 xn−2 . For 9), α2 xn−1 β2 xn−2 xn−1 = β2 xn−2 xn−1 α2 xn−1 . Since xn−1 α2 = α2 xn−1 and xn−1 β2 = β2 xn−1 , by cancellation, we have α2 β2 xn−1 xn−2 = β2 xn−2 α2 xn−1 . By Lemma 3.2, we have α2 β2 = γxn−2 for some γ ∈ MB n which is a contradiction since xn−2 - α2 β2 . ¤ Lemma 3.8. If xi - γ for some i (1 < i < n − 1), then γ can be written as a product of γ1 and γ2 , where xh |γ1 for all h < i and xj |γ2 for all j > i.

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Proof. Locate a subword xj xh in γ such that h < i < j. If there is no such a subword, we are done, otherwise commute xh and xj (it is possible as |h − j| ≥ 2). Continue the process of locating these subwords and commutation. The process stops as γ has finite length and we have the required γ1 and γ2 . ¤ Proof of Proposition 1.8. The induction starts at n = 4. There are 13 vertices in Γ(SB4 ) and we have a unique major component (the edge x1 − x3 − x1 x3 − x1 ). The vertex set V of Γ(SBn ) can be written in two disjoint sets V1 = {γ : xn−1 - γ} and V2 = {γ : xn−1 |γ}. By induction, there is only one major component in the subgraph on the set V1 and this component contains x1 . If γ ∈ V2 commutes with α ∈ V2 , then by Theorem 1.3, some xi for 2 ≤ i ≤ n − 2 is missing in γ. By Lemma 3.8, γ can be written as γ1 γ2 such that xn−2 - γ1 ∈ V1 and hence there is an edge x1 − xn−1 − γ1 − γ. ¤ References [1] U. Ali, A. Haider, Centralizer of braids and Fibonacci numbers, Utilitas Mathematica 89 (2012) 289–296. [2] D.J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986) 641–659. [3] E. Artin, Theory of braids, Ann. Math. 48 (1947) 101–126. [4] R. Ashraf, B. Berceanu, Simple braids (arXiv:1003.6014v1 [math.GT], 2010). [5] R. Ashraf, B. Berceanu, A. Riasat, Fibonacci numbers and positive braids (arXiv: 1005.1145v1 [math.CO], 2010), Ars Combinatoria (to appear). [6] R. Ashraf, B. Berceanu, A. Riasat, What could be a simple permutation (arXiv: 1007.3869v1 [math.CO], 2010). [7] C. Bates, D. Bondy, S. Perkins, P. Rowley, Commuting involution graphs for symmetric groups, J. Algebra 266 (1) (2003) 133–153. [8] B. Berceanu, Artin algebras and applications in topology (in Romanian), PhD thesis, University of Bucharest, 1995. [9] G. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978) 178–218. [10] D. Bondy, The connectivity of commuting graphs, J. Combin. Theory (Ser. A) 113 (2006) 995–1007. [11] L.A. Bokut, Y. Fong, W.-F. Ke, L.-S. Shiao, Gr¨ obner-Shirshov bases for the braid semigroup, in: Advances in Algebra, World Sci., River Edge, NJ, 2003, pp. 60–72. [12] K.S. Brown, The geometry of rewriting systems: a proof of the Anick-Groves-Squier theorem, in: Algorithms and Classification in Combinatorial Group Theory (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ. 23, Springer, New York, 1992, pp. 137–163. [13] P.M. Cohn, Further Algebra and Applications, Springer-Verlag, London, 2003. [14] D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, W.P. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992. [15] F.A. Garside, The braid group and other groups, Quart. J. Math. Oxford (Ser. 2) 20 (1969) 235–254. [16] A. Iranmanesh, A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, J. Algebra Appl. 7 (1) (2008) 129–146. [17] V.A. Ufnarovskij, Combinatorial and asymptotic methods in algebra, in: Algebra, VI, Encyclopaedia Math. Sci. 57, Springer, Berlin, 1995, pp. 1–196. [18] D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 1996.