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

Abstract

The reduced Burau representation is a one-parameter representation of 𝐵_𝑛, the braid group on 𝑛 strings. Specializing the parameter to nonzero complex number 𝑥 gives a representation 𝛽_𝑛(𝑥): 𝐵_𝑛→𝐺𝐿(ℂ^𝑛−1), which is either irreducible or has an irreducible composition factor 𝛽̂‌_𝑛(𝑥): 𝐵_𝑛→𝐺𝐿(ℂ^𝑛−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.

1. Introduction

Let 𝐵_𝑛 be the braid group on 𝑛 strings. We consider the linear representation of 𝐵_𝑛 called the Burau representation [1], which has a composition factor, the reduced Burau representation𝛽_𝑛(𝑡)∶𝐵_𝑛⟶GL_𝑛−1ℂ𝑡^±1,(1.1) where 𝑡 is an indeterminate. Specializing 𝑡→𝑥, where 𝑥∈ℂ^∗, defines a representation 𝛽_𝑛(𝑥)∶𝐵_𝑛→GL_𝑛−1(ℂ) which is either irreducible or has an irreducible subrepresentation 𝛽̂‌_𝑛(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 of 𝑘irreducible representations of the braid group, 𝐵_𝑛, irreducible? The question was answered in the cases 𝑘=2and 𝑘=3. We prove that the tensor product of an irreducible 𝛽_𝑛(𝑦) or 𝛽̂‌_𝑛(𝑦) with an irreducible 𝛽_𝑛(𝑧) or 𝛽̂‌_𝑛(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 𝐵_𝑛. In other words, we show that the tensor product of an irreducible 𝛽_𝑛(𝑥) or 𝛽̂‌_𝑛(𝑥) with an irreducible 𝛽_𝑛(𝑦) or 𝛽̂‌_𝑛(𝑦) with an irreducible 𝛽_𝑛(𝑧) or 𝛽̂‌_𝑛(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:𝛽_𝑛𝑥₁⊗𝛽_𝑛𝑥₂⊗⋯⊗𝛽_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1,𝛽_𝑛𝑥₁⊗𝛽̂‌_𝑛𝑥₂⊗⋯⊗𝛽_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−1⊗ℂ^𝑛−2⊗⋯⊗ℂ^𝑛−1,⋮𝛽̂‌_𝑛𝑥₁⊗𝛽̂‌_𝑛𝑥₂⊗⋯⊗𝛽̂‌_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−2⊗ℂ^𝑛−2⊗⋯⊗ℂ^𝑛−2.(1.2)

Our main result is that, for 𝑛≥4, the above representations are irreducible if (−1)^𝑙𝑤_𝑙≠(−1)^𝑡𝑤_𝑡^±1, for 𝑙,𝑡∈{1,…,𝑘}, where 𝑤_𝑙 and 𝑤_𝑡 are positive words of lengths 𝑙 and 𝑡,respectively, and do not have any 𝑥_𝑖in common.

2. Definitions

The braid group on 𝑛 strings, 𝐵_𝑛, is defined as an abstract group with 𝑛−1 generators 𝜎_𝑖 (𝑖=1,2,…,𝑛−1) and relations:(i)𝜎_𝑖𝜎_𝑖+1𝜎_𝑖=𝜎_𝑖+1𝜎_𝑖𝜎_𝑖+1 for 𝑖=1,2,…,𝑛−2,(ii)𝜎_𝑖𝜎_𝑗=𝜎_𝑗𝜎_𝑖if|𝑖−𝑗|≥2.

The generators 𝜎₁,𝜎₂…,𝜎_𝑛−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 𝐵_𝑛 modules and ℂ[𝐵_𝑛] modules will mean the same. We define the following representations of 𝐵_𝑛 by matrices over ℂ[𝑡^±1].

Definition 1. The reduced Burau representation 𝛽_𝑛(𝑡)∶𝐵_𝑛→GL_𝑛−1(ℂ[𝑡^±1]) is given by 𝜎₁(𝑡)=𝛽_𝑛(𝑡)𝜎₁=⎛⎜⎜⎜⎜⎝−𝑡0−1100𝐼_𝑛−3⎞⎟⎟⎟⎟⎠,𝜎_𝑖(𝑡)=𝛽_𝑛(𝑡)𝜎_𝑖=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐼_𝑖−20001−𝑡00−𝑡00−11000𝐼_{𝑛−𝑖−2}⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,for𝑖=2,…,𝑛−2,𝜎_𝑛−1(𝑡)=𝛽_𝑛(𝑡)𝜎_𝑛−1=⎛⎜⎜⎜⎜⎝𝐼_𝑛−3001−𝑡0−𝑡⎞⎟⎟⎟⎟⎠.(2.1)

Definition 2. 𝜏_𝑛=𝜎₁𝜎₂⋯𝜎_𝑛−1and 𝜏_𝑛(𝑡)=𝜎₁(𝑡)𝜎₂(𝑡)⋯𝜎_𝑛−1(𝑡).

Direct calculations show that𝜏_𝑛(𝑡)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝑡𝑡²−𝑡³⋯(−𝑡)^𝑛−2(−𝑡)^𝑛−1−100⋯000−10⋯0000−1⋯00000⋯−10⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2.2)

We identify ℂ^𝑛−1 with (𝑛−1)×1 column vectors, we let 𝑒₁,𝑒₂,…,𝑒_𝑛−1 denote the standard basis for ℂ^𝑛−1, and we consider matrices to act by left multiplication on column vectors.

Definition 3. If 𝑟=𝑎₁𝑒₁+𝑎₂𝑒₂+⋯+𝑎_𝑛−1𝑒_𝑛−1∈ℂ^𝑛−1, the support of 𝑟, denoted by supp(𝑟), is the set {𝑒_𝑖∣𝑎_𝑖≠0}. If 𝑠=∑𝑎_𝑖𝑗(𝑒_𝑖⊗𝑒_𝑗)∈ℂ^𝑛−1⊗ℂ^𝑛−1, the support of 𝑠, also denoted by supp(𝑠), is the set {𝑒_𝑖⊗𝑒_𝑗∣𝑎_𝑖𝑗≠0}, and 𝑎_𝑖𝑗 is called the coefficient of𝑒_𝑖⊗𝑒_𝑗 in 𝑠.

Definition 4. For 𝑖=1,2,…,𝑛−1,𝑣_𝑖(𝑡)=𝑒_𝑖−𝜎_𝑖(𝑡)(𝑒_𝑖)=(𝐼−𝜎_𝑖(𝑡))(𝑒_𝑖).

3. Preliminaries

Lemma 3.1. (a) 𝜎_𝑖(𝑡)(𝑣_𝑖−1(𝑡))=𝑣_𝑖−1(𝑡)−𝑣_𝑖(𝑡) for 2≤𝑖≤𝑛−1,
𝜎_𝑖(𝑡)(𝑣_𝑖(𝑡))=−𝑡𝑣_𝑖(𝑡) for 1≤𝑖≤𝑛−1,
𝜎_𝑖(𝑡)(𝑣_𝑖+1(𝑡))=−𝑡𝑣_𝑖(𝑡)+𝑣_𝑖+1(𝑡) for 1≤𝑖≤𝑛−2,
𝜎_𝑖(𝑡)(𝑣_𝑗(𝑡))=𝑣_𝑗(𝑡) for 1≤𝑖,  𝑗≤𝑛−1,  |𝑖−𝑗|≥2,
(b) 𝜏_𝑛(𝑣_𝑖(𝑡))=−𝑣_𝑖+1(𝑡) for 1≤𝑖≤𝑛−2.

Note that the above lemma remains true for any specialization 𝑡→𝑦, where 𝑦∈ℂ^∗.

Notation 1. Fix nonzero complex numbers 𝑥₁,…,𝑥_𝑘. Let 𝜎∈𝐵_𝑛and 𝑢=𝑢₁⊗⋯⊗𝑢_𝑘∈ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1, then we write 𝜎(𝑢₁⊗⋯⊗𝑢_𝑘)=𝜎(𝑥₁)(𝑢₁)⊗⋯⊗𝜎(𝑥_𝑘)(𝑢_𝑘).

Notation 2. Given 𝑥₁,…,𝑥_𝑘∈ℂ^∗, by a positive word of length 𝑙, we mean a word that is written as a product of 𝑥₁,…,𝑥_𝑘, where the number of 𝑥_𝑖s involved is 𝑙, and their exponents are ones. Each 𝑥_𝑖in 𝑤_𝑙appears exactly once. We denote the word by 𝑤_𝑙. As an example, we write 𝑤₃=𝑥₁𝑥₃𝑥₄to stand for a word of length 3.

Lemma 3.2. 𝜎^𝑛_𝑖⎛⎜⎜⎝𝑒_𝑖⊗𝑒_𝑖⊗⋯⊗𝑒_{𝑖𝑘-times}⎞⎟⎟⎠=⎧⎪⎪⎨⎪⎪⎩𝑒_𝑖−1+𝑥^𝑛₁1+𝑥₁𝑣_𝑖𝑥₁⊗⋯⊗𝑒_𝑖−1+𝑥^𝑛_𝑘1+𝑥_𝑘𝑣_𝑖𝑥_𝑖𝑘if𝑛isodd,𝑒_𝑖−1−𝑥^𝑛₁1+𝑥₁𝑣_𝑖𝑥₁⊗⋯⊗𝑒_𝑖−1−𝑥^𝑛_𝑘1+𝑥_𝑘𝑣_𝑖𝑥_𝑘if𝑛iseven.(3.1)

Lemma 3.3. Let 𝛽_𝑛(𝑦)∶𝐵_𝑛→GL(ℂ^𝑛−1) be a specialization of the Burau representation making ℂ^𝑛−1 into a 𝐵_𝑛-module, where 𝑛≥3, then (a)let 𝐴 be the kernel of the homomorphism ℂ[𝐵_𝑛]→ℂ induced by 𝜎_𝑖→1 (the augmentation ideal), then 𝐴ℂ^𝑛−1 is equal to the ℂ-vector space spanned by 𝑣₁(𝑦),…,𝑣_𝑛−1(𝑦),(b)if 𝑀 is a nonzero 𝐵_𝑛-submodule of ℂ^𝑛−1, then 𝐴ℂ^𝑛−1⊆𝑀. Hence, 𝐴ℂ^𝑛−1 is the unique minimal nonzero 𝐵_𝑛-module of ℂ^𝑛−1,(c)if 𝑦 is not a root of 𝑝(𝑡)=𝑡^𝑛−1+𝑡^𝑛−2+⋯+𝑡+1, then 𝐴ℂ^𝑛−1=ℂ^𝑛−1, and 𝛽_𝑛(𝑦) is irreducible. If 𝑦 is a root of 𝑝(𝑡), then dim_𝐶(𝐴ℂ^𝑛−1)=𝑛−2.

Hence, 𝐴ℂ^𝑛−1 is its unique minimal nonzero 𝐵_𝑛-submodule. Of course, 𝐴ℂ^𝑛−1=ℂ^𝑛−1 when 𝛽_𝑛(𝑦) is irreducible, but when 𝛽_𝑛(𝑦) is reducible, 𝐴ℂ^𝑛−1 is the subrepresentation 𝛽̂‌_𝑛(𝑦).

The main technical result is Proposition 4.1, which says that if (−1)^𝑙𝑤_𝑙≠(−1)^𝑡𝑤_𝑡^±1,for 𝑙,𝑡∈{1,…,𝑘},where 𝑤_𝑙and 𝑤_𝑡are positive words that do not have any 𝑥_𝑖in common, then 𝐴ℂ^𝑛−1⊗𝐴ℂ^𝑛−1⊗⋯⊗𝐴ℂ^𝑛−1_{𝑘-times}is the unique minimal nonzero 𝐵_𝑛-module of ℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−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 of 𝑘 Irreducible Representations (𝑘≥2)

We obtain a result concerning the irreducibility of the tensor product of 𝑘irreducible representations of the braid group, 𝐵_𝑛, 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. Let 𝑀 be a nonzero 𝐵_𝑛 submodule of ℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1_{𝑘-times} under the action of 𝛽_𝑛(𝑥₁)⊗𝛽_𝑛(𝑥₂)⊗⋯⊗𝛽_𝑛(𝑥_𝑘)∶𝐵_𝑛→GL(ℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1), where 𝑛≥3. Suppose that for 𝑙,𝑡∈{1,…,𝑘} one has (−1)^𝑙𝑤_𝑙≠(−1)^𝑡𝑤_𝑡^±1𝑤_𝑙and𝑤_𝑡donothaveany𝑥_𝑖incommon,(4.1) then, 𝑀 contains all 𝑣_𝑡1(𝑥₁)⊗𝑣_𝑡2(𝑥₂)⊗⋯⊗𝑣_𝑡𝑘(𝑥_𝑘) for 1≤𝑡₁,𝑡₂,…,𝑡_𝑘≤𝑛−1, and 𝑀 contains 𝐴ℂ^𝑛−1⊗𝐴ℂ^𝑛−1⊗⋯⊗𝐴ℂ^𝑛−1_{𝑘-times}, where the action of 𝐵_𝑛 on the 𝑖th factor is induced by 𝛽_𝑛(𝑥_𝑖).

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 exists 𝑚∈𝑀 such that 𝑒₁⊗𝑒₁⊗⋯⊗𝑒_{1𝑘-times}or 𝑒₂⊗𝑒₂⊗⋯⊗𝑒_{2𝑘-times}∈supp(𝑚).

Proof of Claim 1. When 𝑘=2, we set 𝐽=𝑝∣𝑒_𝑝⊗𝑒_𝑞or𝑒_𝑞⊗𝑒_𝑝∈supp(𝑚)forsome𝑚∈𝑀andsome𝑞.(4.2)
First, we show that 1∈𝐽. Second, we let 𝑝=𝜎₃⋯𝜎_𝑘, where 𝑘=min{𝑝∣𝑒_𝑝∈supp(𝑣)∪supp(𝑤)}. Using some recursive argument, we show that there is an element 𝑚or 𝑝(𝑚)of the form 𝑚=𝑎𝑒₁⊗𝑒₂+𝑏𝑒₂⊗𝑒₁+𝑊,(4.3) 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 −𝑎𝑧−by=coefficientof𝑒₁⊗𝑒₁in𝜎₂(𝑚),𝑎𝑧²−𝑧+𝑏𝑦²−𝑦=coefficientof𝑒₁⊗𝑒₁in𝜎₂²(𝑚).(4.4) The determinant det⎛⎜⎜⎝−𝑧−𝑦𝑧²−𝑧𝑦²−𝑦⎞⎟⎟⎠=𝑦𝑧(𝑧−𝑦),(4.5) 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 case 𝑘≥2, we follow the same argument as above. The 2×2above is generalized to a (2^𝑘−2)×(2^𝑘−2) matrix 𝐴_𝑘 whose rows 𝑅₁,𝑅₂,…,𝑅_2𝑘−2 are given by𝑅₁=(−1)^𝑙𝑤_𝑙,1≤𝑙≤𝑘−1,𝑅_𝑖=𝑓_𝑖𝑤_𝑙,1≤𝑙≤𝑘−1,𝑤_𝑙arethesamepositivewordsin𝑅₁,(4.6) for 𝑖=2,…,2^𝑘−2.

Here, 𝑓_𝑖(𝑥_𝑙)=𝛼_𝑖(𝑥_𝑙) for 𝑖=1,…,2^𝑘−2,and 𝑤_𝑙s are all positive words of length 𝑙. Using the hypothesis, the determinant of the matrix 𝐴_𝑘 is nonzero, and this gives that one of the coefficients in 𝜎^𝑖₂(𝑚) has 𝑒₁⊗𝑒₁⊗⋯⊗𝑒_{1𝑘-times}∈supp(𝑚) for 𝑖=1,2,…,2^𝑘−2.

Claim 2. Suppose that 𝑒_𝑖⊗𝑒_𝑖⊗⋯⊗𝑒_{𝑖𝑘-times}∈supp(𝑚) for some 𝑚∈𝑀, then 𝑣_𝑖⊗𝑣_𝑖⊗⋯⊗𝑣_{𝑖𝑘-times}∈𝑀.

Proof of Claim 2. 𝜎_𝑖−1∏^𝑘−1_𝑡=1𝜎_𝑖−(−1)^𝑡𝑤_𝑡𝑒_𝑖⊗𝑒_𝑖⊗⋯⊗𝑒_𝑖_{𝑘-times}=∏^𝑘_𝑡=2𝑤_𝑡−(−1)^𝑡∏−𝑤₁^{2𝑘−1−1}(𝑣_𝑖⊗⋯⊗𝑣_𝑖)_{𝑘-times},𝜎_𝑖−1^𝑘−1_𝑡=1𝜎_𝑖−(−1)^𝑡𝑤_𝑡𝑒_𝑖1⊗𝑒_𝑖2⊗⋯⊗𝑒_𝑖𝑘=0if𝑖₁,…,𝑖_𝑘≠(𝑖,…,𝑖).(4.7) Here, 𝑤_𝑡represents all words of length 𝑡. By our hypothesis, the proof is done.

Claim 3. There exists at least one (𝑠₁,𝑠₂,…,𝑠_𝑘) such that 𝑣_𝑠1⊗𝑣_𝑠2⊗⋯⊗𝑣_𝑠𝑘∈𝑀, where 𝑠₁,𝑠₂,…,𝑠_𝑘∈{1,2} and (𝑠₁,𝑠₂,…,𝑠_𝑘) is neither (1,1,…,1) nor (2,2,…,2).

Proof of Claim 3. Knowing that 𝑣₁⊗𝑣₁⊗⋯⊗𝑣_{1𝑘-times}∈𝑀, then 𝜎₂(𝑣₁⊗𝑣₁⊗⋯⊗𝑣₁)∈𝑀. It follows that 𝑣₁−𝑣₂⊗𝑣₁−𝑣₂⊗⋯⊗𝑣₁−𝑣₂∈𝑀.(4.8)
The proof is completed by applying 𝜎₁ repeatedly to the expression above.

Claim 4. All the tensors 𝑣_𝑠1⊗𝑣_𝑠2⊗⋯⊗𝑣_𝑠𝑘∈𝑀 for all 𝑠₁,𝑠₂,…,𝑠_𝑘∈{1,2}.

Proof of Claim 4. By applying 𝜎₂ to the tensors obtained from Claim 3 and then applying 𝜎₁ repeatedly, the proof is done.

Claim 5. For 3≤𝑖≤𝑛−2, all tensors of the form 𝑣_𝑖⊗𝑣₁⊗⋯⊗𝑣_{1𝑘-1times} and 𝑣_𝑖⊗𝑣₂⊗⋯⊗𝑣_{2𝑘-1times} and those obtained by permuting the indices are in 𝑀.

Proof of Claim 5. Knowing that 𝜎_𝑖+1𝑣_𝑖𝑥₁⊗𝑣₁𝑥₂⊗⋯⊗𝑣₁𝑥_𝑘=𝑣_𝑖𝑥₁−𝑣_𝑖+1𝑥₁⊗𝑣₁𝑥₂⊗⋯⊗𝑣₁𝑥_𝑘,𝜎_𝑖+1𝑣_𝑖𝑥₁⊗𝑣₂𝑥₂⊗⋯⊗𝑣₂𝑥_𝑘=𝑣_𝑖(𝑥)−𝑣_𝑖+1(𝑥)⊗𝑣₂𝑥₂⊗⋯⊗𝑣₂𝑥_𝑘,(4.9) we apply induction on 𝑖, and the proof is finished.

Claim 6. All tensors 𝑣_𝑖⊗𝑣_𝑗⊗𝑣₁⊗⋯⊗𝑣_{1𝑘-2times} and those obtained by permuting the indices are in 𝑀 for 𝑖,𝑗≥3.

Proof of Claim 6. This is done by induction on 𝑗 and using Claim 5.

Claim 7. All tensors of the form 𝑣_𝑖⊗𝑣_𝑗⊗𝑣_ℎ⊗𝑣₁⊗𝑣₁⊗⋯⊗𝑣_{1𝑘-3times} and those obtained by permuting the indices are in 𝑀.

Proof of Claim 7. Applying induction on ℎ and using Claim 6, the proof is completed.

Claim 8. All tensors 𝑣₁⊗𝑣_𝑠1⊗𝑣_𝑠2⊗⋯⊗𝑣_𝑠𝑘−1 are in 𝑀 for all 1≤𝑠₁,𝑠₂,…,𝑠_𝑘−1≤𝑛−1.

Proof of Claim 8. This is done by induction on 𝑠_𝑘−1 and using the previous claims.

Claim 9. 𝑣_𝑠1⊗𝑣_𝑠2⊗⋯⊗𝑣_𝑠𝑘∈𝑀 for all 1≤𝑠₁,𝑠₂,…,𝑠_𝑘−1≤𝑛−1.

Proof of Claim 9. By Claim 8, we have that all the tensors 𝑣₁⊗𝑣_𝑠1⊗𝑣_𝑠2⊗⋯⊗𝑣_𝑠𝑘−1are in 𝑀, Consider then 𝜏_𝑛𝑣₁⊗𝑣_𝑠1⊗𝑣_𝑠2⊗⋯⊗𝑣_𝑠𝑘−1∈𝑀.(4.10) Knowing that 𝜏_𝑛(𝑣_𝑖(𝑡))=−𝑣_𝑖+1(𝑡), the proof is completed.

We now get our main theorem.

Theorem 4.2. Consider the tensor products of the irreducible representations 𝛽_𝑛𝑥₁⊗𝛽_𝑛𝑥₂⊗⋯⊗𝛽_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1,where𝑝𝑥₁≠0,𝑝𝑥₂≠0,…,𝑝𝑥_𝑘≠0,𝛽_𝑛𝑥₁⊗𝛽̂‌_𝑛𝑥₂⊗⋯⊗𝛽_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−1⊗ℂ^𝑛−2⊗⋯⊗ℂ^𝑛−1,where𝑝𝑥₁≠0,𝑝𝑥₂=0,…,𝑝𝑥_𝑘≠0,⋮𝛽̂‌_𝑛𝑥₁⊗𝛽̂‌_𝑛𝑥₂⊗⋯⊗𝛽̂‌_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−2⊗ℂ^𝑛−2⊗⋯⊗ℂ^𝑛−2,where𝑝𝑥₁=0,𝑝𝑥₂=0,…,𝑝𝑥_𝑘=0,(4.11) where 𝛽_𝑛(𝑥)∶𝐵_𝑛→GL(ℂ^𝑛−1) and 𝛽̂‌_𝑛(𝑥)∶𝐵_𝑛→GL(𝐴ℂ^𝑛−1)=GL(ℂ^𝑛−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 (−1)^𝑙𝑤_𝑙≠(−1)^𝑡𝑤_𝑡^±1𝑤_𝑙and𝑤_𝑡donothaveany𝑥_𝑖incommon,(4.12) then the above representations are irreducible.

Proof. The proof is along the same lines as in the special cases 𝑘=2 and 𝑘=3. All of the above representations are subrepresentations of 𝛽_𝑛𝑥₁⊗𝛽_𝑛𝑥₂⊗⋯⊗𝛽_𝑛𝑥_𝑘∶𝐵_𝑛⟶GLℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1.(4.13)
By Proposition 4.1, 𝐴ℂ^𝑛−1⊗𝐴ℂ^𝑛−1⊗⋯⊗𝐴ℂ^𝑛−1_{𝑘-times} is the unique minimal nonzero 𝐵_𝑛 module of ℂ^𝑛−1⊗ℂ^𝑛−1⊗⋯⊗ℂ^𝑛−1_{𝑘-times}. In particular, it is an irreducible 𝐵_𝑛-module. By Lemma 3.3, the first factor 𝐴ℂ^𝑛−1 corresponds to one of the representations 𝛽_𝑛(𝑥₁) or 𝛽̂‌_𝑛(𝑥₁), the second factor 𝐴ℂ^𝑛−1 corresponds to one of the representations 𝛽_𝑛(𝑥₂) or 𝛽̂‌_𝑛(𝑥₂), and so on according to whether or not 𝑥₁, 𝑥₂,…𝑥_𝑘 are roots of 𝑝(𝑡). Hence, all the above representations can be identified with the 𝐵_𝑛-module 𝐴ℂ^𝑛−1⊗𝐴ℂ^𝑛−1⊗⋯⊗𝐴ℂ^𝑛−1_{𝑘-times}, so they are irreducible.