# Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

Authors: Mohammad N. Abdulrahim; Wiaam M. Zeid
Journal: ISRN Algebra
 (2011)

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## Abstract

The reduced Burau representation is a one-parameter representation ofBn, the braid group on n strings. Specializing the parameter to nonzero complex number x gives a representation βn(x): Bn→GL(ℂn−1), which is either irreducible or has an irreducible composition factor β^n(x): Bn→GL(ℂn−2). In our paper, we let k≥2, and we determine a sufficient condition for the irreducibility of the tensor product of k irreducible Burau representations. This is a generalization of our previous work concerning the cases k=2 and k=3.


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## Body

## 1. Introduction

LetBn be the braid group on n strings. We consider the linear representation of Bn called the Burau representation [1], which has a composition factor, the reduced Burau representation(1.1)βn(t):Bn⟶GLn-1(C[t±1]),
where t is an indeterminate. Specializing t→x, where x∈ℂ*, defines a representation βn(x):Bn→GLn-1(ℂ) which is either irreducible or has an irreducible subrepresentation β̂n(x) of degree n-2. For more details, see [2, page 286].In our paper, we consider the following question: for which values of the parameters is the tensor product ofkirreducible representations of the braid group, Bn, irreducible? The question was answered in the cases k=2and k=3. We prove that the tensor product of an irreducible βn(y) or β̂n(y) with an irreducible βn(z) or β̂n(z) is irreducible if and only if y≠z±1. For more details, see [3]. We also consider the case k=3and find a sufficient condition that guarantees the irreducibility of three irreducible representations of Bn. In other words, we show that the tensor product of an irreducible βn(x) or β̂n(x) with an irreducible βn(y) or β̂n(y) with an irreducible βn(z) or β̂n(z) is irreducible if x≠y±1,  x≠z±1,  y≠z±1, x+yz≠0,  y+xz≠0,  z+xy≠0,  and xyz≠-1. We fall short of finding a necessary and sufficient condition in the case k=3(see [4]).In our paper, we generalize the results obtained in [3, 4] and find a sufficient condition that guarantees the irreducibility of the tensor product of kirreducible representations of the braid group, where k≥2. Therefore, our paper is concerned with the tensor product of the following irreducible representations:(1.2)βn(x1)⊗βn(x2)⊗⋯⊗βn(xk):Bn⟶GL(Cn-1⊗Cn-1⊗⋯⊗Cn-1),βn(x1)⊗β̂n(x2)⊗⋯⊗βn(xk):Bn⟶GL(Cn-1⊗Cn-2⊗⋯⊗Cn-1),⋮β̂n(x1)⊗β̂n(x2)⊗⋯⊗β̂n(xk):Bn⟶GL(Cn-2⊗Cn-2⊗⋯⊗Cn-2).Our main result is that, forn≥4, the above representations are irreducible if (-1)lwl≠(-1)twt±1, for l,t∈{1,…,k}, where wl and wt are positive words of lengths l and t,respectively, and do not have any xiin common.


## 2. Definitions

The braid group onn strings, Bn, is defined as an abstract group with n-1 generators σi (i=1,2,…,n-1) and relations:(i)
σiσi+1σi=σi+1σiσi+1 for i=1,2,…,n-2,(ii)
σiσj=σjσiif|i-j|≥2.The generatorsσ1,σ2…,σn-1 are called the standard generators. Let t be an indeterminate, and let ℂ[t±1] be a Laurent polynomial ring over the complex numbers. All modules are ℂ-vector spaces, so Bn modules and ℂ[Bn] modules will mean the same. We define the following representations of Bn by matrices over ℂ[t±1].Definition 1.
The reduced Burau representationβn(t):Bn→GLn-1(ℂ[t±1]) is given by
(2.1)σ1(t)=βn(t)(σ1)=(-t0-1100In-3),σi(t)=βn(t)(σi)=(Ii-20001-t00-t00-11000In-i-2),fori=2,…,n-2,σn-1(t)=βn(t)(σn-1)=(In-3001-t0-t).Definition 2.
τn=σ1σ2⋯σn-1and  τn(t)=σ1(t)σ2(t)⋯σn-1(t).Direct calculations show that(2.2)τn(t)=(-tt2-t3⋯(-t)n-2(-t)n-1-100⋯000-10⋯0000-1⋯00000⋯-10).We identifyℂn-1 with (n-1)×1 column vectors, we let e1,e2,…,en-1 denote the standard basis for ℂn-1, and we consider matrices to act by left multiplication on column vectors.Definition 3.
Ifr=a1e1+a2e2+⋯+an-1en-1∈ℂn-1, the support of r, denoted by supp(r), is the set {ei∣ai≠0}. If s=∑aij(ei⊗ej)∈ℂn-1⊗ℂn-1, the support of s, also denoted by supp(s), is the set {ei⊗ej∣aij≠0}, and aij is called the coefficient ofei⊗ej in s.Definition 4.
Fori=1,2,…,n-1,vi(t)=ei-σi(t)(ei)=(I-σi(t))(ei).


## 3. Preliminaries

Lemma 3.1.
(a)σi(t)(vi-1(t))=vi-1(t)-vi(t) for 2≤i≤n-1,
σi(t)(vi(t))=-tvi(t) for 1≤i≤n-1,
σi(t)(vi+1(t))=-tvi(t)+vi+1(t) for 1≤i≤n-2,
σi(t)(vj(t))=vj(t) for 1≤i,  j≤n-1,  |i-j|≥2,
(b)τn(vi(t))=-vi+1(t)  for 1≤i≤n-2.Note that the above lemma remains true for any specializationt→y, where y∈ℂ*.Notation 1.
Fix nonzero complex numbersx1,…,xk. Let σ∈Bnand u=u1⊗⋯⊗uk∈ℂn-1⊗⋯⊗ℂn-1, then we write σ(u1⊗⋯⊗uk)=σ(x1)(u1)⊗⋯⊗σ(xk)(uk).Notation 2.
Givenx1,…,xk∈ℂ*, by a positive word of length l, we mean a word that is written as a product of x1,…,xk, where the number of xis involved is l, and their exponents are ones. Each xiin wlappears exactly once. We denote the word by wl. As an example, we write w3=x1x3x4to stand for a word of length 3.Lemma 3.2.
(3.1)σin(ei⊗ei⊗⋯⊗ei︸k-times)={(ei-(1+x1n1+x1)vi(x1))⊗⋯⊗(ei-(1+xkn1+xk)vi(xik))ifnis odd,(ei-(1-x1n1+x1)vi(x1))⊗⋯⊗(ei-(1-xkn1+xk)vi(xk))ifnis even.Lemma 3.3.
Letβn(y):Bn→GL(ℂn-1) be a specialization of the Burau representation making ℂn-1 into a Bn-module, where n≥3, then (a)
letA be the kernel of the homomorphism ℂ[Bn]→ℂ induced by σi→1 (the augmentation ideal), then Aℂn-1 is equal to the ℂ-vector space spanned by v1(y),…,vn-1(y),(b)
ifM is a nonzero Bn-submodule of ℂn-1, then Aℂn-1⊆M. Hence, Aℂn-1 is the unique minimal nonzero Bn-module of ℂn-1,(c)
ify is not a root of p(t)=tn-1+tn-2+⋯+t+1, then Aℂn-1=ℂn-1, and βn(y) is irreducible. If y is a root of p(t), then dimC(Aℂn-1)=n-2.Hence,Aℂn-1 is its unique minimal nonzero Bn-submodule. Of course, Aℂn-1=ℂn-1 when βn(y) is irreducible, but when βn(y) is reducible, Aℂn-1 is the subrepresentation β̂n(y).The main technical result is Proposition4.1, which says that if (-1)lwl≠(-1)twt±1,for l,t∈{1,…,k},where wland wtare positive words that do not have any xiin common, then Aℂn-1⊗Aℂn-1⊗⋯⊗Aℂn-1︸k-timesis the unique minimal nonzero Bn-module of ℂn-1⊗ℂn-1⊗⋯⊗ℂn-1︸k-times. This implies the irreducibility of the tensor products under the above conditions. We will mainly follow the argument presented in [3, 4]. However, new techniques in the proof are needed to generalize our computations.


## 4. Tensor Product ofk Irreducible Representations (k≥2)

We obtain a result concerning the irreducibility of the tensor product ofkirreducible representations of the braid group, Bn, where k≥2. We state our proposition and give an outline of a proof that goes along the same lines as in the cases k=2and k=3. However, a more general adequate proof is required here, which could be easily verified in the cases k=2and k=3by simply returning back to our previous work in [3, 4]. Most of the formulas and equations in the proof can be verified using mathematical induction and possibly by performing direct computations as well.Proposition 4.1.
LetM be a nonzero Bn submodule of ℂn-1⊗ℂn-1⊗⋯⊗ℂn-1︸k-times under the action of βn(x1)⊗βn(x2)⊗⋯⊗βn(xk):Bn→GL(ℂn-1⊗ℂn-1⊗⋯⊗ℂn-1), where n≥3. Suppose that for l,t∈{1,…,k} one has
(4.1)(-1)lwl≠(-1)twt±1(wlandwtdonothaveanyxiincommon),
then, M contains all vt1(x1)⊗vt2(x2)⊗⋯⊗vtk(xk) for 1≤t1,t2,…,tk≤n-1, and M contains Aℂn-1⊗Aℂn-1⊗⋯⊗Aℂn-1︸k-times, where the action of Bn on the ith factor is induced by βn(xi).Proof.
The steps of the proof are similar to those in [3, 4]. But still, we need to generalize our computations in the general case k≥2.Claim 1.
There existsm∈M such that e1⊗e1⊗⋯⊗e1︸k-timesor e2⊗e2⊗⋯⊗e2︸k-times∈supp(m).Proof of Claim1.
Whenk=2, we set
(4.2)J={p∣ep⊗eqoreq⊗ep∈supp⁡(m)forsomem∈Mandsomeq}.
First, we show that1∈J. Second, we let p=σ3⋯σk, where k=min{p∣ep∈supp(v)∪supp(w)}. Using some recursive argument, we show that there is an element mor p(m)of the form
(4.3)m=a(e1⊗e2)+b(e2⊗e1)+W,
where e1⊗e1,e1⊗e2,e2⊗e1∉supp(W), and at least one of a,b is nonzero. If e2⊗e2∈supp(W), we are done. If e2⊗e2∉supp(W), then
(4.4)-az-by=coefficientofe1⊗e1inσ2(m),a(z2-z)+b(y2-y)=coefficientofe1⊗e1in(σ2)2(m).
The determinant
(4.5)det⁡(-z-yz2-zy2-y)=yz(z-y),
is nonzero, since y≠z. Then one of -az-by,a(z2-z)+b(y2-y) is nonzero, and one of σ2(m), (σ2)2(m) has e1⊗e1 in its support. For more details, see [3].As for the general casek≥2, we follow the same argument as above. The 2×2above is generalized to a (2k-2)×(2k-2) matrix Ak whose rows R1,R2,…,R2k-2 are given by(4.6)R1={(-1)lwl,1≤l≤k-1},Ri={{fi(wl),1≤l≤k-1,wlarethesamepositivewordsinR1},
for i=2,…,2k-2.Here,fi(xl)=αi(xl) for i=1,…,2k-2,and wls are all positive words of length l. Using the hypothesis, the determinant of the matrix Ak is nonzero, and this gives that one of the coefficients in σ2i(m) has e1⊗e1⊗⋯⊗e1︸k-times∈supp(m) for i=1,2,…,2k-2.Claim 2.
Suppose thatei⊗ei⊗⋯⊗ei︸k-times∈supp(m) for some m∈M, then vi⊗vi⊗⋯⊗vi︸k-times∈M.Proof of Claim2.
(4.7)(σi-1)∏t=1k-1(σi-(-1)twt)(ei⊗ei⊗⋯⊗ei)︸k-times=∏t=2k(wt-(-1)t)∏(-w1)2k-1-1(vi⊗⋯⊗vi)︸k-times,(σi-1)∏t=1k-1(σi-(-1)twt)(ei1⊗ei2⊗⋯⊗eik)=0if(i1,…,ik)≠(i,…,i).
Here, wtrepresents all words of length t. By our hypothesis, the proof is done.Claim 3.
There exists at least one(s1,s2,…,sk) such that vs1⊗vs2⊗⋯⊗vsk∈M, where s1,s2,…,sk∈{1,2} and (s1,s2,…,sk) is neither (1,1,…,1) nor (2,2,…,2).Proof of Claim3.
Knowing thatv1⊗v1⊗⋯⊗v1︸k-times∈M, then σ2(v1⊗v1⊗⋯⊗v1)∈M. It follows that
(4.8)(v1-v2)⊗(v1-v2)⊗⋯⊗(v1-v2)∈M.
The proof is completed by applyingσ1 repeatedly to the expression above.Claim 4.
All the tensorsvs1⊗vs2⊗⋯⊗vsk∈M for all s1,s2,…,sk∈{1,2}.Proof of Claim4.
By applyingσ2 to the tensors obtained from Claim 3 and then applying σ1 repeatedly, the proof is done.Claim 5.
For3≤i≤n-2, all tensors of the form vi⊗v1⊗⋯⊗v1︸k-1 times and vi⊗v2⊗⋯⊗v2︸k-1 times and those obtained by permuting the indices are in M.Proof of Claim5.
Knowing that(4.9)σi+1(vi(x1)⊗v1(x2)⊗⋯⊗v1(xk))=(vi(x1)-vi+1(x1))⊗v1(x2)⊗⋯⊗v1(xk),σi+1(vi(x1)⊗v2(x2)⊗⋯⊗v2(xk))=(vi(x)-vi+1(x))⊗v2(x2)⊗⋯⊗v2(xk),
we apply induction on i, and the proof is finished.Claim 6.
All tensorsvi⊗vj⊗v1⊗⋯⊗v1︸k-2 times and those obtained by permuting the indices are in M for i,j≥3.Proof of Claim6.
This is done by induction onj and using Claim 5.Claim 7.
All tensors of the formvi⊗vj⊗vh⊗v1⊗v1⊗⋯⊗v1︸k-3 times and those obtained by permuting the indices are in M.Proof of Claim7.
Applying induction onh and using Claim 6, the proof is completed.Claim 8.
All tensorsv1⊗vs1⊗vs2⊗⋯⊗vsk-1 are in M for all 1≤s1,s2,…,sk-1≤n-1.Proof of Claim8.
This is done by induction onsk-1 and using the previous claims.Claim 9.
vs1⊗vs2⊗⋯⊗vsk∈M for all 1≤s1,s2,…,sk-1≤n-1.Proof of Claim9.
By Claim8, we have that all the tensors v1⊗vs1⊗vs2⊗⋯⊗vsk-1are in M, Consider then
(4.10)τn(v1⊗vs1⊗vs2⊗⋯⊗vsk-1)∈M.
Knowing that τn(vi(t))=-vi+1(t), the proof is completed.We now get our main theorem.Theorem 4.2.
Consider the tensor products of the irreducible representations(4.11)βn(x1)⊗βn(x2)⊗⋯⊗βn(xk):Bn⟶GL(Cn-1⊗Cn-1⊗⋯⊗Cn-1),wherep(x1)≠0,p(x2)≠0,…,p(xk)≠0,βn(x1)⊗β̂n(x2)⊗⋯⊗βn(xk):Bn⟶GL(Cn-1⊗Cn-2⊗⋯⊗Cn-1),wherep(x1)≠0,p(x2)=0,…,p(xk)≠0,⋮β̂n(x1)⊗β̂n(x2)⊗⋯⊗β̂n(xk):Bn⟶GL(Cn-2⊗Cn-2⊗⋯⊗Cn-2),wherep(x1)=0,p(x2)=0,…,p(xk)=0,
where βn(x):Bn→GL(ℂn-1) and β̂n(x):Bn→GL(Aℂn-1)=GL(ℂn-2) denote a specialization of the reduced Burau representation and the irreducible subrepresentation of Lemma 3.3(b), respectively. If, for l,t∈{1,…,k}, one has that
(4.12)(-1)lwl≠(-1)twt±1(wlandwtdonothaveanyxiincommon),
then the above representations are irreducible.Proof.
The proof is along the same lines as in the special casesk=2 and k=3. All of the above representations are subrepresentations of
(4.13)βn(x1)⊗βn(x2)⊗⋯⊗βn(xk):Bn⟶GL(Cn-1⊗Cn-1⊗⋯⊗Cn-1).
By Proposition4.1, Aℂn-1⊗Aℂn-1⊗⋯⊗Aℂn-1︸k-times is the unique minimal nonzero Bn module of ℂn-1⊗ℂn-1⊗⋯⊗ℂn-1︸k-times. In particular, it is an irreducible Bn-module. By Lemma 3.3, the first factor Aℂn-1 corresponds to one of the representations βn(x1) or β̂n(x1), the second factor Aℂn-1 corresponds to one of the representations βn(x2) or β̂n(x2), and so on according to whether or not x1, x2,…xk are roots of p(t). Hence, all the above representations can be identified with the Bn-module Aℂn-1⊗Aℂn-1⊗⋯⊗Aℂn-1︸k-times, so they are irreducible.



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