The Interplay between Linear Representations of the Braid Group

International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 16186,9pages

doi:10.1155/2007/16186

Research Article

The Interplay between Linear Representations of the Braid Group

Mohammad N. Abdulrahim and Nibal H. Kassem

Received 19 May 2007; Accepted 10 July 2007

Recommended by Howard E. Bell

We consider Wada’s representation as a twisted version of the standard action of the braid group,Bn, on the free group withngenerators. Constructing a free group,Gnm, of rank nm, we compose Cohen’s mapBn→Bnmand the embeddingBnm→Aut(Gnm) via Wada’s

map. We prove that the composition factors of the obtained representation are one copy of Burau representation andm−1 copies of the standard representation after changing the parameterttotk _{in the definitions of the Burau and standard representations. This}

is a generalization of our previous result concerning the standard Artin representation of the braid group.

Copyright © 2007 M. N. Abdulrahim and N. H. Kassem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-erly cited.

1. Introduction

There are many kinds of representations ofBn, the braid group onnstrings. The earliest

was the Artin representation, which is an embeddingBn→Aut(Fn), the automorphism

group of a free group onn generators [1, page 25]. A certain type of representation, introduced by F. R. Cohen and studied by him and others, is the mapBn→Bnmwhich is

defined on geometric braids by replacing each string withmstrings [2, page 208]. InSection 2of this paper, we present an infinite series of representations generalizing the standard Artin representation, which were discovered by M. Wada [3]. More pre-cisely, for an arbitrary nonzero integerk, the automorphism corresponding to the braid generatorσitakesxitoxikxi+1xi−k;xi+1toxi, and fixes all other free generators.

InSection 3, we compose Cohen’s map with Wada’s representation and we get a lin-ear representation of degreenmwhich has a subrepresentation isomorphic to the Burau representation, and the quotient is isomorphic to the direct sum ofm−1 copies of the standard representation, which was studied by Sysoeva [5]. This is done after we change the indeterminatettotk _{in the definitions of the Burau and standard representations.}

As a corollary, by lettingk=1, we get our previous result concerning the standard Artin representation of the braid group. For more details, see [6].

2. Notation and preliminaries

The braid group onnstrings, Bn, is an abstract group which has a presentation with

generators

σ1,. . .,σn−1 (2.1)

and defining relations

σiσi+1σi=σi+1σiσi+1 fori=1, 2,. . .,n−2,

σiσj=σjσi if|i−j| ≥2. (2.2)

The generatorsσ1,. . .,σn−1are called the standard generators ofBn. Lettbe an

indetermi-nate and letC[t±1_{] represent the Laurent polynomial ring over complex numbers.}

Definition 2.1. The Burau representationβn(t) :Bn→GLn(C[t±1]) is defined by

βn(t)

σi

=

⎛ ⎜ ⎜ ⎜ ⎝

Ii−1 0 0

0 1−t t

1 0 0

0 0 In−i−1

⎞ ⎟ ⎟ ⎟

⎠ fori=1,. . .,n−1. (2.3)

The standard representationγn(t) :Bn→GLn(C[t±1]) is defined by

γn(t)

σi

=

⎛ ⎜ ⎜ ⎜ ⎝

Ii−1 0 0

0 0 t

1 0 0

0 0 In−i−1 ⎞ ⎟ ⎟ ⎟

⎠ fori=1,. . .,n−1. (2.4)

For more details about the standard representation, see [5].

There is a well-known standard representation (due to Artin) of groupBnin group

Aut(Fn) of automorphisms of the free group Fn generated by x1,. . .,xn. The

automor-phismσicorresponding to the braid generatorσitakesxi→xixi+1xi−1;xi+1→xi, and fixes

all other free generators.

A twisted version of the standard action of the braid group on the free group is Wada’s representation; thus we have the following definition.

takes

xi−→xikxi+1xi−k, xi+1−→xi,

xj−→xj forj=i,i+ 1.

(2.5)

Definition 2.3 [7, page 104]. LetGbe an arbitrary group and letZGbe the group ring of

Gwith respect to the ring of integersZ. A mappingD:ZG→ZGis said to be a derivative if and only if

(1)D(f+h)=D f+Dhand

(2)D(f h)=(D f)(h) +f(Dh) (product rule) for all f andhinZG.

Here, is the augmentation homomorphism: ZG→Z defined by (g∈Gngg)=

g∈Gng.

LetFnbe a free group of rankn, with free basisx1,. . .,xn. We define for j=1, 2,. . .,n

the free derivatives on the groupZFnby

∂ ∂xj

x1 μ1···x

r

μr

= r

i=1

iδμi,jxμ11···x

(1/2)(i−1)

μi ,

∂ ∂xj

agg

=ag∂g

∂xj, g∈Fn,ag∈Z,

(2.6)

wherei= ±1 andδi,jis the Kronecker symbol.

The following properties hold true. (i)∂xi/∂xj=δi,j.

(ii)∂x−1

i /∂xj= −δi,jxi−1.

(iii) (∂/∂xj)(uv)=(∂u/∂xj)(v) +u(∂v/∂xj)u,v∈ZFn.

Note that ifv∈Fn, then(v)=1. For simplicity, we denote∂/∂xjbydj.

Using the Magnus representation, the automorphismσiunder Wada’s representation

is mapped onto then×nmatrix [φ((∂/∂xr)σi(xj))] which differs from the identity only

by a 2×2 block with the top left corner in the (i,i)th place. More precisely,

σi(t)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Ii−1 0 0

0 1−t

k _tk

1 0 0

0 0 In−i−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟

⎠ fori=1, 2,. . .,n−1. (2.7)

Ifφis an arbitrary homomorphism acting onFndefined asφ(xi)=t, thenφ(yi)=tk

fori=1,. . .,n. LetGnφdenote the image ofGnunderφ.

Under Wada’s representation, the action of the generators ofBnon the free groupFn

induces an action on the free subgroupGn. That is, we have a faithful representation of Bnas a subgroup of Aut(Gn).

Lemma 2.4. Under Wada’s representation, the action ofσion the basis ofGn, namely,{y1,. . ., yn}, is given by

yi−→yiyi+1yi−1, yi+1−→yi,

yr−→yr, r=i,i+ 1.

(2.8)

Proof. σi(yi)=σi(xik)=(σi(xi))k=xikxi+1xi−kxikxi+1xi−k···xikxi+1xi−k=xikxi+1kxi−k= yiyi+1yi−1.

The action ofσion the other generators follows easily.

UsingLemma 2.4and the Magnus representation ofBnas a subgroup of Aut(Gn), the

automorphismσiis mapped onto then×nmatrix [φ((∂/∂yr)σi(ys))]. Direct

computa-tions show that it is the same matrix as in (2.7). Therefore, we get the following corollary.

Corollary 2.5. Under Wada’s representation, then×nmatrices obtained by lettingBnact

onFnor onGnare exactly the same.

Proof. This follows easily fromLemma 2.4and the fact that we have definedφ(yi)=tk.

3. Automorphisms ofGnm

Definition 3.1 [2, page 208]. The Cohen representation is the mapBn→Bnmdefined as

follows:

σi−→1×σi=

σmiσmi+1···σmi+m−1

σmi−1σmi···σmi+m−2

···σmi−m+1σmi−m+2···σmi

.

(3.1)

Here, 1×σiis the braid obtained by replacing each string of the geometric braid,σi,

withmparallel strings. Cohen called 1×σia tensor product.

Puttingk=1 in the definition of Wada’s map, we get the result in [6], which asserts that by composing Cohen’s map with Artin’s representation of the braid group, we get a linear representation:Bn→Bnm→GLnm(Z[t±1]) which has a subrepresentation

isomor-phic to the Burau representation, and the quotient is isomorisomor-phic to the direct sum of

m−1 copies of the standard representation, which was studied by Sysoeva [5].

Given the free generatorsx1,. . .,xnm, we letyi=xikfori=1,. . .,nm. We takeGnmto be

the free group generated byy1,. . .,ynm.

Let τi be the image of the braid generator σi of Bn under the Cohen map. Using Lemma 2.4, there is an induced action ofτi on the free subgroupGnm. As inSection 2,

we show that the (nm)×(nm) matrix obtained by lettingτias act onFnmwith generators x1,. . .,xnm is exactly the same as that obtained by havingτi act onGnm with generators x1k,. . .,xnmkinstead. Therefore, we get the following theorem.

Theorem 3.2. The action of the image of the generator ofBnunder Cohen’s map, namely,τi,

onFnmgives an (nm)×(nm) matrix which is the same as the one obtained under the action

ofτion the free subgroupGnm.

Proof. Let

τi=σmiσmi+1···σmi+m−1

σmi−1σmi···σmi+m−2

···σmi−m+1σmi−m+2···σmi. (3.2)

Let us see the action ofτionFnmwith generatorsx1,. . .,xnm.

It is clear that we need to see the action ofτiespecially on the 2melements, namely,

xmi−m+1,xmi−m+2,. . .,xmi,xmi+1,xmi+2,. . .,xmi+m. (3.3)

As for the other elements, the action ofτiis trivial. Direct computations show that

τi

xmi−m+s

=xmi−m+1k···xmik

xmi+s

xmi−m+1k···xmik −1

fors=1,. . .,m. (3.4)

Also, we have that

τixmi+s=xmi+s−m fors=1,. . .,m. (3.5)

The action ofτion the free subgroupGnmwith generatorsy1,. . .,ynm, whereyj=xjk

forj=1,. . .,nm, is given by

τiymi−m+s=ymi−m+1···ymiymi+symi−m+1···ymi−1 fors=1,. . .,m. (3.6)

Also, we have that

τiymi+s=ymi+s−m fors=1,. . .,m. (3.7)

Next, we apply Magnus representation to get the matrices corresponding toτi, namely,

[φ((∂/∂xr)τi(xs))] and [φ((∂/∂yr)τi(ys))]. Using Fox derivatives and having defined φ(xj)=tandφ(yj)=tk for j=1,. . .,nm, we get that the matrices are the same. To see

this, we make some computations.

For fixed values ofiandm, we denoteφ((∂/∂yr)τi(ymi−m+s)) orφ((∂/∂xr)τi(xmi−m+s))

equal. More precisely, we have that

dmi−m+1

τi

ymi−m+s

=1−tk_{, d} mi−m+2

τi

ymi−m+s

=tk_−t2k_,

dmi−m+3

τi

ymi−m+s

=t2k_−t3k_{,. . .,d} mi

τi

ymi−m+s

=t(m−1)k_−tmk_. (3.8)

For 2≤s≤m, we have

dmi+1τiymi−m+s= ··· =dmi+s−1τiymi−m+s=0. (3.9)

Also, we have that for 1≤s≤m

dmi+s

τi

ymi−m+s

=tmk. (3.10)

Ifs≤m−1, then

dmi+s+1

τi

ymi−m+s

= ··· =dmi+m

τi

ymi−m+s

=0. (3.11)

As for the elementsymi+s, we have that

dpτiymi+s=δp,mi+s−m (3.12)

(δi,jis the Kronecker symbol).

Notice that form=1, we getCorollary 2.5.

Throughout our work, we will then treat the generators ofBnas automorphisms of the

free groupGnmwith generatorsy1,. . .,ynm, whereyi=xikrather than automorphisms of Fnm.

Next, we proceed as in [6] by choosing elementszi,jofGnm, each of which is a word in

theseyi’s. More precisely, for 1≤i≤mand 1≤j≤nwe definezi,jas follows:

zi,j=y1+m j−my2+m j−m. . . ym j−i+1. (3.13)

It is then clear that for fixed choices of a positive integer,m, and an integeri: 1≤i≤m, the length ofzi,jism−i+ 1. In other words, the generators{zi,j}are defined as follows:

z1,1=y1···ym, z2,1=y1···ym−1, . . ., zm,1=y1,

z1,2=y1+m···y2m, z2,2=y1+m···y2m−1, . . ., zm,2=y1+m,

..

. ... ...

z1,n=y1+(n−1)m···ynm, z2,n=y1+(n−1)m···ynm−1, . . ., zm,n=y1+(n−1)m.

(3.14)

Letτrbe the automorphism onGnmthat corresponds toτrwhich is the image of the

braid generatorσrofBnunder the Cohen map. When there is no danger of confusion, we

will still denote the automorphismτrbyτr.

UsingLemma 2.4inSection 2of our work and [6, Theorem 3.1, page 172], we easily get the following theorem.

Theorem 3.4. For 1≤r≤n−1 and 1≤i≤m, the action ofτron the basis{zi,j}ofGnmis

given by

(1)zi,r→z1,rzi,r+1z1,r−1,

(2)zi,r+1→zi,r,

(3)zi,j→zi,j, 1≤j≤n(j=r,r+ 1).

Letφ(zi,j)=tkfor 1≤i≤mand 1≤j≤n. LetDi,j=φ(∂/∂zi,j). Now to find the linear

representation

Bn−→Bnm−→GL(nm,Z)[t±1], (3.15)

we determine the Jacobian matrix of the image of the braid generatorσr under Cohen

map, namely the automorphismτron the groupGnm. But first, we give an order to the

generators ofGnmas follows:

z1,1,z1,2,. . .,z1,n,z2,1,z2,2,. . .,z2,n,. . .,zm,1,zm,2,. . .,zm,n. (3.16)

Then we define the Jacobian matrix as follows:

Jτr= ⎛ ⎜ ⎜ ⎝

D1,1τrz1,1 ··· Dm,nτrz1,1

..

. ...

D1,1

τr

zm,n

··· Dm,n

τr

zm,n

⎞ ⎟ ⎟

⎠. (3.17)

We now prove our main theorem.

Theorem 3.5. The linear representation obtained by composing the Cohen representation

with Wada’s representation has a subrepresentation isomorphic to the Burau representation ofBn, and the quotient is isomorphic to the direct sum ofm−1 copies of the standard

rep-resentation ofBn after changing the parametertto tk in the definitions of the Burau and

standard representations. More precisely,

σr−→ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

βntkσr 0 ··· 0

γntkσr ...

. .. 0

γntkσr ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. (3.18)

Proof. UsingDefinition 2.3for free derivatives andTheorem 3.4, we get for 1≤i≤m

D1,r

τr

zi,r

=1−tk_{, D} i,r+1

τr

zi,r

Also notice that

Di,rτrzi,r+1=1 (3.20)

(hereφ(zi,j)=tk).

We take this subrepresentation as the one specified by the basis{z1,1,. . .,z1,n}. The

direct summands of the quotient are generated by the images of{zi,1,. . .,zi,n}fori=2, . . .,m. In other words, the Jacobian matrix ofτris given by

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 0 ··· 0

0 . .. 1

1−tk _tk

1 0

1 ..

. . .. ...

1

1−tk _{0 0 t}k

0 0 1 0

1 . .. ..

. ... ... 1 ...

1−tk _{0 0 t}k

0 0 1 0

1

. .. 0

0 ··· 0 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (3.21)

RecallingDefinition 2.1, we have then proved our theorem.

Notice that, fork=1, we get the result that was proved in [6].

References

[1] E. Artin, The Collected Papers of Emil Artin, Addison-Wesley, Reading, Mass, USA, 1965. [2] F. R. Cohen, “Artin’s braid groups and classical homotopy theory,” in Combinatorial Methods in

Topology and Algebraic Geometry (Rochester, NY, 1982), vol. 44 of Contemporary Mathematics,

pp. 207–220, American Mathematical Society, Providence, RI, USA, 1985.

[3] V. Shpilrain, “Representing braids by automorphisms,” International Journal of Algebra and

Computation, vol. 11, no. 6, pp. 773–777, 2001.

[4] M. N. Abdulrahim, “Generalizations of the standard Artin representation are unitary,”

Interna-tional Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 8, pp. 1321–1326, 2005.

[6] M. N. Abdulrahim, “On the composition of the Burau representation and the natural mapBn→

Bnk,” Journal of Algebra and Its Applications, vol. 2, no. 2, pp. 169–175, 2003.

[7] J. S. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, no. 82, Princeton University Press, Princeton, NJ, USA, 1974.

Mohammad N. Abdulrahim: Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut 11072809, Lebanon

Email address:[email protected]

Nibal H. Kassem: Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut 11072809, Lebanon