Wada s Representations of the. Pure Braid Group of High Degree

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1 Theoretical Mathematics & Applications, vol2, no1, 2012, ISSN: (print), (online) International Scientific Press, 2012 Wada s Representations of the Pure Braid Group of High Degree Ihsan A Daakur 1 and Mohammad N Abdulrahim 2 Abstract We consider the representation obtained by composing the embeddingmapofthepurebraidgroupp n P n+k andwada srepresentation of degree n+k to get a linear representation P n GL n+k (C[t ±1 1,,t±1 n+k ]), whose composition factors are to be determined A similar work was done in a previous work in the case of the Gassner representation of P n Mathematics Subject Classification: 20F36 Keywords: pure braid group, Wada s representation 1 Introduction The braid group on n strands, denoted by B n, is defined as an abstract group with generators σ 1,σ 2,,σ n 1 and relations σ i σ j = σ j σ i, i j 2, 1 i, j n 1; σ i σ i+1 σ i = σ i+1 σ i σ i+1 for 1 i n 2 1 Department of Mathematics, Beirut Arab University, PO Box: , Beirut, Lebanon, iad101@bauedulb 2 Department of Mathematics, Beirut Arab University, PO Box: , Beirut, Lebanon, mna@bauedulb Article Info: Received : December 28, 2011 Revised : January 19, 2012 Published online : March 5, 2012

2 118 Wada s Representations of the Pure Braid Group of High Degree The pure braid group, P n, is a normal subgroup of the braid group, B n, on n strings It has many linear representations One of them is Wada s representation, which is an embedding P n Aut(F n ), the automorphism group of a free group on n generators In [1], Abdulrahim has constructed an embedding of the pure braid group P n P n+k and composed it with the Gassner representation of P n+k to get a linear representation P n GL n+k (C[t ±1 1,,t ±1 n,,t ±1 n+k ]), where the composition factors were completely determined In our work, we consider Wada s representation instead of the Gassner representation, where σ i, takes x i x i x 1 i+1 x i, x i+1 x i ; and fixes all other free generators Our main theorem is similar to that obtained in [1], where the composition factors of the representation obtained by composing the embedding map and Wada s representation are to be determined However, for the sake of our work, the embedding map P n P n+k has to be defined in a different way, where a generator of P n is mapped to another generator of P n+k, rather than to a product of generators of P n+k as in [1] 2 Preliminaries Definition 21 Let F n be a free group of rank n, with free basis x 1,,x n We define for j = 1,2,,n the free derivatives on the group ZF n by (i) x i xj = δ i,j, (ii) x 1 i = δ xj i,j x 1 i, (iii) x j (uv) = u v ɛ(v)+u u,v ZF xj xj Here δ i,j is the Kronecker symbol For simplicity, we denote x j by d j Definition 22 The pure braid group, P n, is defined as the kernel of the homomorphism B n S n, defined by σ i (i,i+1),1 i n 1 It has the following generators: A i,,j j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 σ j 1, 1 i < j n

3 I Daakur and M Abdulrahim 119 Definition 23 Wada s representation is defined as the representation of theautomorphism correspondingtothe braidgeneratorσ i, takes x i x i x 1 i+1 x i, x i+1 x i and fixes all other free generators It is easy to see that the inverse σ 1 i, takes x i x i+1, x i+1 x 1 i+1 x ix 1 i+1 ; and fixes all other generators For more details, see [3] 3 Main Results We determine the action of the automorphisms A i,j on the generators of the free group F n We then define an embedding of the pure braid group P n P n+k and compose the embedding map and Wada s representation of the pure braid group P n+k 31 Action of the automorphisms on the free group Lemma 31 As automorphisms of the free group F n, the generators, A i,j, act on the free group F n as follows: (i) A i,j (x i ) = x i x 1 x 1 (ii) A i,j (x j ) = x i x 1 (iii) A i,j (x r ) = x r (iv) A i,j (x r ) = (x i x 1 x j )x 1 r (x x 1 ) if 1 r < i or j < r n if i < r < j Proof We have that A i,,j j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 σ j 1, 1 i < j n We need to consider A i,j as left automorphisms acting on the generators of F n from the left To prove (i): σ 1 j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 σ j 1 (x i )

4 120 Wada s Representations of the Pure Braid Group of High Degree j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 (x i ) j 1 σ 1 j 2 σ 1 i+1 σ i(σ i (x i )) j 1 σ 1 j 2 σ 1 i+1 σ i(x i x 1 j 1 σ 1 j 2 σ 1 i+1 (x ix 1 i+1 x ix 1 i x i x 1 j 1 σ 1 j 2 σ 1 i+1 (x ix 1 i+1 x ix 1 j 1 σ 1 j 2 σ 1 i+2 (x ix 1 i+2 x ix 1 i+2 x i) j 1 σ 1 j 2 σ 1 i+3 (x ix 1 i+3 x ix 1 i+3 x i) j 1 σ 1 j 2 (x ix 1 j 2 x ix 1 j 2 x i) j 1 (x ix 1 j 1 x ix 1 j 1 x i) = x i x 1 x 1 To prove (ii): σ 1 j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 σ j 1 (x j ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 (x j 1 ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 3 (x j 2 ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 4 (x j 3 ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 (x i+2 ) j 1 σ 1 j 2 σ 1 i+1 σ2 i(x i+1 ) j 1 σ 1 j 2 σ 1 i+1 σ i(x i ) j 1 σ 1 j 2 σ 1 i+1 (x ix 1 j 1 σ 1 j 2 σ 1 i+2 (x ix 1 i+2 x i) j 1 σ 1 j 2 σ 1 i+3 (x ix 1 i+3 x i) j 1 σ 1 j 2 σ 1 i+4 (x ix 1 i+4 x i) j 1 σ 1 j 2 (x ix 1 j 2 x i) j 1 (x ix 1 j 1 x i) = x i x 1 To prove (iii): Since r > j, that is, the smallest possible value of r is j+1, then A i,j (x r ) = x r for j < r n Now, since r < i, that is, the greatest possible value of r is i 1, then A i,j (x r ) = x r for 1 r < i

5 I Daakur and M Abdulrahim 121 To prove (iv): Since i < r < j, the largest index of x, namely r, is j 1 and the smallest index of x, namely r, is i 1 Then σ 1 j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ j 2 σ j 1 (x r ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ r 1 σ r (x r ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ r 1 (x r x 1 r+1x r ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 σ r 2 (x r 1 x 1 r+1x r 1 ) j 1 σ 1 j 2 σ 1 i+1 σ2 iσ i+1 (x i+2 x 1 r+1x i+2 ) j 1 σ 1 j 2 σ 1 i+1 σ2 i(x i+1 x 1 r+1x i+1 ) j 1 σ 1 j 2 σ 1 i+1 σ i(x i x 1 r+1x i ) j 1 σ 1 j 2 σ 1 i+1 (x ix 1 i+1 x ix 1 r+1x i x 1 j 1 σ 1 j 2 σ 1 i+2 (x ix 1 i+2 x ix 1 r+1x i x 1 i+2 x i) j 1 σ 1 j 2 σ 1 i+3 (x ix 1 i+3 x ix 1 r+1x i x 1 i+3 x i) j 1 σ 1 j 2 σ 1 r (x i x 1 r x i x 1 r+1x i x 1 r x i ) j 1 σ 1 j 2 σ 1 r+1(x i x 1 r+1x i x r+1 x 1 r x r+1 x i x 1 r+1x i ) j 1 σ 1 j 2 σ 1 r+2(x i x 1 r+2x i x r+2 x 1 r x r+2 x i x 1 r+2x i ) j 1 σ 1 j 2 (x ix 1 j 2 x ix j 2 x 1 r x j 2 x i x 1 j 2 x i) j 1 (x ix 1 j 1 x ix j 1 x 1 r x j 1 x i x 1 j 1 x i) = x i x 1 x j x 1 r x x 1 = (x i x 1 x j )x 1 r (x x 1 ) 32 The embedding P n P n+k In [1], the embedding of the pure braid group was defined in a way that the generators A 1,j were mapped to a product of generators of P n+k and other generators A i,j to A i+k,j+k Whereas in our work, we require that the generator of P n is to be mapped to another generator of P n+k More precisely, we define the following map: Ψ : P n P n+k

6 122 Wada s Representations of the Pure Braid Group of High Degree as Ψ(A i,j ) = { A 1,j+k, i = 1 and 2 j n A i+k,j+k, 2 i < j n, where Ψ(A i,j ) is a generator of P n+k, that is an automorphism of F n+k whose action on the free generator x 1,x 2,,x n,,x n+k is defined in Lemma 31 We compose the map above with the embedding P n+k Aut(F n+k ) The image of the generators under this embedding is treated as left automorphisms of the free froup F n+k As a basis for the free group F n+k, we take the following generators: y 1 = x 1, y 2 = x k+2, y 3 = x k+3,,y n = x k+n, y n+1 = x 2, y n+2 = x 3,,y n+k = x k+1 Using the action of the automorphism, σ i, on the basis of F n+k, we have the following lemmas about the images of the generators of P n, namely Ψ(A i,j ) for 1 i < j n Lemma 32 For 1 < j n, the action of the images of the generators, A 1,j on the basis of F n+k is given by (i) Ψ(A 1,j )(y 1 ) = y 1 (y 1 j y 1 y 1 j y 1 ), (ii) Ψ(A 1,j )(y j ) = y 1 y 1 j y 1, (iii)ψ(a 1,j )(y r ) = y r for j < r n, (iv) Ψ(A 1,j )(y r ) = (y 1 y 1 j y 1 y j )yr 1 (y j y 1 y 1 j y 1 ) for 1 < r < j, (v) Ψ(A 1,j )(y n+r ) = (y 1 y 1 j y 1 y j )yn+r(y 1 j y 1 y 1 j y 1 ) for 1 r k Proof Since the image of A 1,j under Ψ is a generator of P n+k namely A 1,j+k, it suffices only to prove (v) We have that Ψ(A 1,j ) = A 1,j+k j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1σ 2 2 σ 3 σ k σ k+1 σ k+2 σ j+k 1 To prove (v): Let y n+r = x r+1, for 1 r k Then 1 r < j +k 1 and we have that σ 1 j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1σ 2 2 σ 3 σ k σ k+1 σ k+2 σ j+k 1 (x r+1 ) j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1σ 2 2 σ 3 σ k σ k+1 σ k+2 σ r σ r+1 (x r+1 )

7 I Daakur and M Abdulrahim 123 j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1σ 2 2 σ 3 σ k σ k+1 σ k+2 σ r (x r+1 x 1 r+2x r+1 ) j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1σ 2 2 σ 3 σ k σ k+1 σ k+2 σ r 1 (x r x 1 r+2x r ) j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1σ 2 2 σ 3 σ k σ k+1 σ k+2 σ r 2 (x r 1 x 1 r+2x r 1 ) j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ1(x 2 2 x 1 r+2x 2 ) j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 σ 1 (x 1 x 1 r+2x 1 ) j+k 1 σ 1 j+k 2 σ 1 3 σ2 1 (x 1 x 1 2 x 1 x 1 r+2x 1 x 1 2 x 1 ) j+k 1 σ 1 j+k 2 σ 1 3 (x 1 x 1 3 x 1 x 1 r+2x 1 x 1 3 x 1 ) j+k 1 σ 1 j+k 2 σ 1 4 (x 1 x 1 4 x 1 x 1 r+2x 1 x 1 4 x 1 ) j+k 1 σ 1 j+k 2 σ 1 r+2σr+1(x 1 1 x 1 r+1x 1 x 1 r+2x 1 x 1 r+1x 1 ) j+k 1 σ 1 j+k 2 σ 1 r+2(x 1 x 1 r+2x 1 x r+2 x 1 r+1x r+2 x 1 x 1 r+2x 1 ) j+k 1 σ 1 j+k 2 σ 1 r+3(x 1 x 1 r+3x 1 x r+3 x 1 r+1x r+3 x 1 x 1 r+3x 1 ) j+k 1 (x 1x 1 j+k 1 x 1x j+k 1 x 1 r+1x j+k 1 x 1 x 1 j+k 1 x 1) = x 1 x 1 j+k x 1x j+k x 1 r+1x j+k x 1 x 1 j+k x 1 = (y 1 y 1 j y 1 y j )y 1 n+r(y j y 1 y 1 j y 1 ) For 1 < i < j n, we have that Ψ(A i,j ) = A i+k,j+k Acting on the generators of F n+k, namely, x 1,,x n+k subject to the rules stated in Lemma 31, we easily verify the following lemma Lemma 33 For 1 < i < j n, the action of the images of the generators A i,j on the basis of F n+k is as follows: (i) Ψ(A i,j )(y i ) = y i (y 1 j y i y 1 j y i ) (ii) Ψ(A i,j )(y j ) = y i y 1 j y i (iii) Ψ(A i,j )(y r ) = y r j < r n (iv)ψ(a i,j )(y r ) = (y i y 1 j y i y j )yr 1 (y j y i y 1 j y i ) (v) Ψ(A i,j )(y n+r ) = y n+r for i < r < j for 1 r < i or for 1 r k Proof As in Lemma 32, we only need to prove (v): Let y n+r = x r+1, for 1 r k Since r k, that is, the greatest possible value of r + 1 is k+1, it follows that Ψ(A i,j )(y n+r ) = Ψ(A i,j )(x r+1 ) = A i+k,j+k (x r+1 ) = x r+1 = y n+r

8 124 Wada s Representations of the Pure Braid Group of High Degree Let φbe a homomorphism from F n+k to (C ) n+k defined by φ(y i ) = t i, for 1 i n + k Let D i = φ y i Our objective now is to determine the Jacobian matrix of the image of the generator A i,j under the map, namely the automorphism Ψ(A i,j ) on the free group F n+k defined in Lemma 32 and Lemma 33, so that we can find the linear representation obtained by composing the map P n P n+k with Wada s representation By intuition, the order of the generators of F n+k is: y 1,,y n,y n+1,,y n+k Consider Ψ(A i,j ), the image of A i,j under the map P n P n+k, and call it A i,j for simplicity Then we define the jacobian matrix as follows: J(A i,j ) = D 1 (A i,j (y 1 )) D n+k (A i,j (y 1 )) D 1 (A i,j (y n+k )) D n+k (A i,j (y n+k )) 119] The construction used here is the Magnus representation of P n+k [2, p115- We now prove our main theorem Theorem 34 By composing the embedding P n P n+k with Wada s representation of P n+k, we get a linear representation of degree n+k whose one of the composition factors is isomorphic to Wada s representation of P n and the other is a diagonal representation The matrix that corresponds to the image of A i,j has the following form: [ γ(a i,j ) 0 where γ(a i,j ) is the image of A i,j under Wada s representation of degree n and M k is a diagonal representation whose diagonal entries are all ones in the case 1 < i n and t 2 1t 1 n+r when i = 1 and 1 r k Proof From Lemma 32 and Lemma 33, we easily verify that statements (i), (ii), (iii) and (iv) coincidewith thedefinitionoftheimage of A i,j underthe M k ],

9 I Daakur and M Abdulrahim 125 Wada s representation of P n specified by the basis {y 1,,y n } (See Lemma 31) Furthermore, statement (v) in Lemma 32 asserts that Ψ(A 1,j )(y n+r ) = (y 1 y 1 j y 1 y j )y 1 n+r(y j y 1 y 1 j y 1 ), for any 1 r k, which in turn implies that D n+r (Ψ(A 1,j )(y n+r )) = t 2 1t 1 n+r Statement (v) in Lemma 33 asserts that for 1 < i < j n, we have that Ψ(A i,j )(y n+r ) = y n+r for 1 r k, which implies that This completes the proof D n+r (Ψ(A i,j )(y n+r )) = 1 4 Conclusion In our work, we have completely determined the composition factors of the representation obtained by composing the embedding map P n P n+k and Wada s representation of the pure braid group P n+k References [1] MN Abdulrahim, On embeddings of pure braid groups P n P n+k, Int J Appl Math, 16(1), (2004), 1-12 [2] JS Birman, Braids, Links and Mapping Class Groups, Annals of Mathematical Studies, vol 82, Princeton University Press, New Jersey, 1975 [3] M Wada, Group Invaraints of Links, Topolgy, 31(1), (1992),