Pure Braid Group Representations of Low Degree

Transcription

1 International Journal of Algebra, Vol 5, 2011, no 21, Pure Braid Group Representations of Low Degree Mohammad N Abdulrahim and Ghenwa H Abboud Department of Mathematics Beirut Arab University PO Box , Beirut, Lebanon mna@bauedulb gha009@bauedulb Abstract We determine the conditions under which we extend the irreducible complex specialization of the reduced Gassner representation of the pure braid group, namely G n (z) :P n Gl n 1 (C) to an irreducible representation P n+1 Gl n 1 (C) This is a continuation to Formanek s work regarding the classification of irreducible representations of the braid group Keywords: pure braid group, Gassner representation, irreducible 1 Introduction The pure braid group, P n, is a normal subgroup of the braid group, B n, on n strings It has a lot of linear representations One of them is the Gassner representation which comes from the embedding P n Aut(F n ), by means of Magnus representation [3, p119] According to Artin, the automorphism corresponding to the braid generator σ i takes x i to x i x i+1 x 1 i,x i+1 to x i, and fixes all other free generators Applying this standard Artin representation to the generator of the pure braid group, we get a representation of the pure braid group by automorphisms Such a representation has a composition factor, the reduced Gassner representation G n (t 1,,t n ) : P n GL n 1 (Z[t ±1 1,,t ±1 n ]), where t 1,,t n are indeterminates Unlike the Burau representation of the braid group, B n, that also arises as a subgroup of the automorphism group of a free group with n generators, there is no explicit representation which is agreed to be the reduced Gassner representation of the pure braid group In our work, we define the reduced Gassner representation up to equivalence Specializing the indeterminates t 1,,t n to nonzero complex numbers z 1,,z n, we get a representation G n (z 1,,z n ) : P n GL n 1 (C) = GL(C n 1 ) which is irreducible if and only if z 1 z n 1 (see [1]) It was shown in [4] that the reduced Burau

2 1012 M N Abdulrahim and G H Abboud representation β n (z) :B n GL n 1 (C) is either irreducible or it has a composition factor of degree n 2 that is irreducible The composition factor is β n (z) :B n GL n 2 (C) =GL(C n 2 ), where β n (z) B n 1 = β n 1 (z) In [2], we get a similar result concerning the pure braid group on 4 strings In that work, we consider the representation G 4 (z 1,,z 4 ):P 4 GL 3 (C) =GL(C 3 ) and we describe the composition factors of G 4 (z 1,,z 4 ) when it is reducible In our paper, we generalize the result obtained in [2] Our immediate goal is to describe the composition factors of G n (z 1,,z n ):P n GL n 1 (C) = GL(C n 1 ) when it is reducible We show that if z 1 1 and z n 1 then G n (z 1,,z n ) has a composition factor of degree n 2, namely, Ĝ n (z 1,,z n ) which is an extension of the irreducible representation G n 1 (z 1,,z n 1 ) to P n In section two, we present the reduced Gassner representation of P n by choosing a free generating set with n generators to be our basis and by applying the Magnus representation as was suggested by J Birman in [3] In section three, we diagonalize the matrix corresponding to some element in P n by an invertible matrix, say T, and conjugate the reduced Gassner representation of P n by the same matrix T Diagonalizing one of the matrices means that the eigen vectors of this diagonal matrix are the standard unit vectors in C n or a linear combination of some of them in case of equal eigenvalues Our objective is to look for invariant subspaces, if exist Then we present our main theorem that shows that if G n (z 1,,z n ):P n GL n 1 (C) =GL(C n 1 ) is reducible, given that z 1 1 and z n 1, then the composition factor Ĝ n (z 1,,z n ):P n GL n 2 (C) =GL(C n 2 ) is an extension of the (irreducible) reduced Gasssner representation G n 1 (z 1,,z n 1 ) to P n and is defined up to equivalence Other observations are made in the cases z 1 =1 and z n =1 2 Definitions Definition 1 The braid group on n strings, denoted by B n, is an abstract group which can be defined via the following presentation: B n =<σ 1,, σ n 1 > and defining relations: σ i σ j = σ j σ i if i j 2 σ i σ i+1 σ i = σ i+1 σ i σ i+1 if 1 i n 2 The generators σ 1,σ 2,, σ n 1 are called the standard generators of B n Definition 2 The pure braid group, denoted by 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 is finitely generated by the elements A ij = σ j 1 σ j 2 σ i+1 σ 2 i σ 1 i+1 σ 1 j 2 σ 1 j 1, 1 i<j n

3 Pure Braid group representations of low degree 1013 These generators have relations among them For more details about the presentations of B n and P n as abstract groups with generators and relations, see [3, pp19 25] Let F n be the free group of rank n, with free basis x 1,, x n In F n, we introduce the free generating set g 1,, g n with g 1 = x 1, g 2 = x 1 x 2,, g n = x 1 x n The action of the braid generators σ i on the basis {g 1,, g n }is defined as follows: σ i : g i g i+1 gi 1 g i 1, i 1 g i g i+1 g 1 i, i =1 g k g k, k i 3 Complex specializations of the reduced gassner representations The action of the pure braid generator, A i,j, on the free basis {g 1,, g n } is defined in the following lemma Lemma 1 Let i<j n and 1 m n Then the action of the pure braid generator is given by the following (i) For i 1, A i,j (g m )= g j g 1 j 1 g ig 1 i 1 g j 1g 1 j g i 1 if m = i g j g 1 j 1 g ig 1 g m i 1 g j 1gj 1 g i 1 gi 1 g m if i<m<j if m<i or m j (ii) A 1,j (g m )= g j g 1 j 1 g 1g j 1 g 1 j if m =1 g j g 1 j 1 g 1g j 1 g 1 j g1 1 g m if 1 <m<j g m if m j Let φ : ZF n Z[t ±1 1,, t ±1 n ], where Z[t ±1 1,, t ±1 n ] is a Laurent polynomial ring in independent variables t 1,, t n φ is defined as φ(g i )=t 1 t i for 1 i n Using Fox derivatives d j defined in [3,p104], we let D j = φd j We then obtain D r (A i,j (g m ))

4 1014 M N Abdulrahim and G H Abboud (t i 1)(t j δ j 1,r δ j,r )+t j δ i,r + t i (1 t j )δ i 1,r, if m = i = (t i 1)(t j δ j 1,r δ j,r )+(t j 1)(δ i,r t i δ i 1,r )+δ m,r, if i<m<j δ m,r, if m<i or m j Here δ i,j is the Kronecker symbol The Jacobian matrix is defined as follows: D 1 (A i,j (g 1 )) D n (A i,j (g 1 )) J(A i,j )= D 1 (A i,j (g n )) D n (A i,j (g n )) Therefore the image of A i,j, denoted by X ij, is given by X ij = I P i,j Q i,j, where P i,j is the column vector defined by and Q i,j (0,, 0, 1,, 1, 0,, 0 }{{}}{{}}{{} i 1 j i +1 is the row vector defined by ) T Q i,j = (0,, 0,t }{{} i (t j 1), 1 t i t j,t i 1, 0,, 0)ifi = j 1 }{{} i 2 (0,, 0,t }{{} i (t j 1), 1 t j, 0,, 0,t }{{} j (1 t i ),t i 1, 0,, 0)ifi j 1 }{{} i 2 j i 2 We have shown that D r (A i,j (g m )) = δ r,m if m j Hence the last row of these n n matrices X i,j is (0,, 0, 1) This immediately implies that the Gassner representation of P n is reducible and is direct sum of trivial representation and a representation of degree n 1, called the reduced Gassner representation of P n and is denoted by G n (t 1,, t n ) Definition 3 The reduced Gassner representation G n (t 1,, t n ):P n GL n 1 (Z[t ±1 1,, t ±1 n ]) is given by G n (A i,j )=I P i,j Q i,j, where P i,j and Q i,j are the same as P i,j and Q i,j after deleting one zero from the last row of P i,j, and one zero from the last column of Q i,j For simplicity, we still denote the matrix corresponding to the generator A i,j by A i,j

5 Pure Braid group representations of low degree Composition Factors of G n (z 1,, z n ) We specialize the indeterminantes t 1,, t n to nonzero complex numbers z 1,, z n For z =(z 1,, z n ) (C ) n, it was proved in [1] that the complex specialization of the reduced Gassner representation of P n, namely G n (z) : P n GL n 1 (C), is irreducible if and only if z 1 z n 1 Notation 1 Let A = A 1,n A 1,n 1 A 1,2 be an element of the pure braid group P n Then we get that A = z 1 z n z 1 z n z 1 z 2 z z 1 z n z 1 z 2 z 3 0 z 1 0 z 1 z n z 1 z n z 1 Theorem 1 Given z =(z 1,, z n ) (C ) n and z 1 z n =1 Let G n (z) : P n GL n 1 (C) be the complex specialization of the reduced Gassner representation (i) If z 1 1and z n 1then the irreducible representation G n 1 (z 1,, z n 1 ): P n 1 GL n 2 (C) extends to a (necessarily irreducible) representation Ĝn(z) : P n GL n 2 (C) (ii) If z 1 =1or z n =1then Ĝn(z) :P n GL n 2 (C) is a composition factor of G n (z) that is completely determined and can be reduced further to a representation of lower degree Proof (i) Let z 1 1,z n 1 and A = A 1,n A 1,n 1 A 1,2 be an element of the pure braid group P n Under the substitution z 1 z n = 1, we get that A = z 1 z 2 z z 1 z 2 z 3 0 z z 1 z n z 1 We then diagonalize this matrix A by the n n matrix T given by

6 1016 M N Abdulrahim and G H Abboud 1 z z 1 z n 1 1 z 1 z z 1 z n 1 1 z 1 z 2 z z T = 1 z n 1 1 z 1 z n z 1 z n It is easy to see that det(t )=( 1) m 1 z 1 [ ] 0, where m = z 1 z n 1 (z n 1) 3n 6 if n is even and m = 3n 7 if n is odd 2 2 Direct computations show that z z T 1 AT = 0 0 z Now we conjugate the reduced Gassner representation G n (z) by the invertible matrix T to get an equivalent representation of degree n 1 We write the matrices as I (T 1 P i,j )(Q i,j T ),where T 1 P i,j = (0,, 0, 1,, 1, 0,, 0) T,i 1 z } z } j i z } i 1 ( z1zn 1 1,, z1zj 1 z1(z2 zj 1 1),,, 1 z 1 1 z 1 1 z 1 z } z1(z2 1) 1 z 1 z } j 2, 1 z1 zn 1 1 z 1 ) T,i=1 and

7 Pure Braid group representations of low degree 1017 Q i,j T = (0,, 0,z i 1, 1 z }{{} i z j,z i (z j 1), 0,, 0, 0), j i =1 }{{} i 1 (0,, 0,z i 1,z }{{} j (1 z i ), 0,, 0, 1 z }{{} j,z i (z j 1), 0,, 0, 0), j i 1 }{{} j i 1 i 1 Since the last entry in Q i,j T is zero, it follows that the last column of the matrix T 1 P i,j Q i,j T is (0,, 0) Hence, the subspace generated by the eigenvector e n 1 is invariant under this representation We thus get an (n 2) (n 2) representation Ĝn(z) :P n GL n 2 (C), defined by A i,j = I P i,j Qi,j, where P i,j and Q i,j are the same as T 1 P i,j and Q i,j T after deleting the last row of T 1 P i,j and the last column of Q i,j T By conjugating the (unreduced) Gassner representation P n 1 GL n 1 (C) by the invertible matrix T defined above, we observe that (1, n 1), (2, n 1),, (n 2,n 1) are zeros Therefore we may delete the last row and last column to obtain an (n 2) (n 2) representation We observe that G n 1 (z 1,,z n 1 ) is exactly the representation Ĝn(z) restricted to the subgroup P n 1 Since z 1 z n = 1 and z n 1, it follows that z 1 z n 1 1 which implies the irreducibility of the (n 2) (n 2) matrix representation of P n 1, but irreducibility on the subgroup P n 1 implies irreducibility on the group P n (ii) Case 1: Let z 1 =1 Under the assumption z 1 z n = 1, we get that z 2 z n =1 Substituting in Definition 3, we see that the first row of A 1,j is (z j, 0,, 0) and that of A i,j is (1, 0,, 0) for i 1 Hence the subspace generated by e 2,e 3,, e n 1 is invariant Therefore, we may delete the first row and the first column to obtain an (n 2) (n 2) representation of P n, namely, Ĝ n (z) :P n GL n 2 (C), defined by A 1,j = I n 2, A i,j = I P i,j Qi,j if i 1, where P i,j and Q i,j are the same as P i,j and Q i,j after deleting the first row of P i,j and the first column of Q i,j Thus, G n (z), which is a representation of degree n 1, has a composition factor of degree (n 2), namely Ĝn(z) By replacing the indeterminates in Ĝn(z), namely, z 2,z 3,,z n by w 1,w 2,,w n 1 respectively, we easily observe that w 1 w 2 w n 1 = 1 We also notice that the matrices A i,j = I P i,j Qi,j (i 1) are exactly the same as the matrices A i 1,j 1 that are obtained under the representation G n 1 (w 1,,w n 1 ): P n 1 GL n 2 (C) after replacing z 2 by w 1,and z n by w n 1 Having

8 1018 M N Abdulrahim and G H Abboud the generators A 1,j = I n 2 and the representation G n 1 (w 1,,w n 1 )is reducible, it follows that the representation Ĝn(z) :P n GL n 2 (C) is reducible Case 2: Let z n =1 We compute A n 1,n under the substitution z 1 z n 1 =1 We see that A n 1,n = ( In z n 1 ) For 1 i n 2, we notice that the last row of A i,n is (0,, 0,z i ) for i n 1 Likewise, the last row of A i,j is (0,, 0, 1) for j n Hence the subspace generated by e 1,e 2,, e n 2 is invariant Therefore, we delete the last row and the last column to obtain an (n 2) (n 2) representation Ĝn(z) :P n GL n 2 (C), defined by A i,n = I n 2, A i,j = I P i,j Qi,j if j n Here, Pi,j and Q i,j are the same as P i,j and Q i,j after deleting the last row of P i,j and the last column of Q i,j Thus, G n (z) of degree (n 1) has a composition factor of degree (n 2), namely Ĝn(z) Here, we observe that Ĝ n (z) is the extension of G n 1 (z) : P n 1 GL n 2 (C) to P n Since z 1 z n 1 =1, and the fact that A i,n = I n 2, it follows that the representation Ĝn(z) :P n GL n 2 (C) is reducible The previous theorem and its proof motivates the following definition Definition 4 Let n 2 and z =(z 1 z n+1 ) If z 1 z n+1 =1, z 1 1 and z n+1 1 then the irreducible representation Ĝ n+1 (z) :P n+1 GL n 1 (C) is defined as Ĝ n+1 (z)(a i,j )=G n (z 1,,z n )(A i,j ), 1 i<j<n+1,

9 Pure Braid group representations of low degree 1019 Ĝ n+1 (z)(a 1,n+1 )= z 1 z n z 1 z n+1 (1 z 2 z n 1 ) z 1 z n+1 (1 z 2 z n 2 ) z 1 z n+1 (1 z 2 ) 0 0 1, Ĝ n+1 (z)(a n,n+1 )= z n z n+1 z n (1 z n+1 ) , and Ĝ n+1 (z)(a i,n+1 )= 1+z n+1 (z i 1) z n+1 1 z i (1 z n+1 ) 0 0 z n+1 (z i 1) z n+1 1 z i (1 z n+1 ) 0 0 z n+1 (z i 1) z n+1 1 z i (1 z n+1 ) 0 0 z n+1 (z i 1) z n+1 z i (1 z n+1 ) 0 0, i n 1 References [1] M Abdulrahim, Complex Specializations of the reduced Gassner representation of the Pure Braid group, Proc of AMS, volume 125, Number 6 (1997),

10 1020 M N Abdulrahim and G H Abboud [2] M Abdulrahim, Gassner Representation of the Pure Braid Group P 4, International Journal of Algebra, volume 3, Number 16 (2009), [3] J S Birman, Braids, Links and Mapping Class Groups, Annals of Mathematical studies, Princeton University Press, New Jersey, volume 82 (1975) [4] E Formanek, Braid group representations of low degree, Proc London Math Soc, volume 73, Number 3 (1996), Received: April, 2011