Table of Contents
ISRN Algebra
Volumeย 2011, Article IDย 104263, 9 pages
http://dx.doi.org/10.5402/2011/104263
Research Article

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

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

Received 17 August 2011; Accepted 5 September 2011

Academic Editors: A. V.ย Kelarev, A.ย Kiliรงman, and A.ย Rapinchuk

Copyright ยฉ 2011 Mohammad N. Abdulrahim and Wiaam M. Zeid. 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 properly cited.

Abstract

The reduced Burau representation is a one-parameter representation of ๐ต๐‘›, the braid group on ๐‘› strings. Specializing the parameter to nonzero complex number ๐‘ฅ gives a representation ๐›ฝ๐‘›(๐‘ฅ): ๐ต๐‘›โ†’๐บ๐ฟ(โ„‚๐‘›โˆ’1), which is either irreducible or has an irreducible composition factor ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฅ): ๐ต๐‘›โ†’๐บ๐ฟ(โ„‚๐‘›โˆ’2). In our paper, we let ๐‘˜โ‰ฅ2, and we determine a sufficient condition for the irreducibility of the tensor product of ๐‘˜ irreducible Burau representations. This is a generalization of our previous work concerning the cases ๐‘˜=2 and ๐‘˜=3.

1. Introduction

Let ๐ต๐‘› be the braid group on ๐‘› strings. We consider the linear representation of ๐ต๐‘› called the Burau representation [1], which has a composition factor, the reduced Burau representation๐›ฝ๐‘›(๐‘ก)โˆถ๐ต๐‘›โŸถGL๐‘›โˆ’1๎€ทโ„‚๎€บ๐‘กยฑ1๎€ป๎€ธ,(1.1) where ๐‘ก is an indeterminate. Specializing ๐‘กโ†’๐‘ฅ, where ๐‘ฅโˆˆโ„‚โˆ—, defines a representation ๐›ฝ๐‘›(๐‘ฅ)โˆถ๐ต๐‘›โ†’GL๐‘›โˆ’1(โ„‚) which is either irreducible or has an irreducible subrepresentation ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฅ) of degree ๐‘›โˆ’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 ๐‘˜irreducible representations of the braid group, ๐ต๐‘›, irreducible? The question was answered in the cases ๐‘˜=2and ๐‘˜=3. We prove that the tensor product of an irreducible ๐›ฝ๐‘›(๐‘ฆ) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฆ) with an irreducible ๐›ฝ๐‘›(๐‘ง) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ง) is irreducible if and only if ๐‘ฆโ‰ ๐‘งยฑ1. For more details, see [3]. We also consider the case ๐‘˜=3and find a sufficient condition that guarantees the irreducibility of three irreducible representations of ๐ต๐‘›. In other words, we show that the tensor product of an irreducible ๐›ฝ๐‘›(๐‘ฅ) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฅ) with an irreducible ๐›ฝ๐‘›(๐‘ฆ) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฆ) with an irreducible ๐›ฝ๐‘›(๐‘ง) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ง) is irreducible if ๐‘ฅโ‰ ๐‘ฆยฑ1,โ€‰โ€‰๐‘ฅโ‰ ๐‘งยฑ1,โ€‰โ€‰๐‘ฆโ‰ ๐‘งยฑ1, ๐‘ฅ+๐‘ฆ๐‘งโ‰ 0,โ€‰โ€‰๐‘ฆ+๐‘ฅ๐‘งโ‰ 0,โ€‰โ€‰๐‘ง+๐‘ฅ๐‘ฆโ‰ 0,โ€‰โ€‰and ๐‘ฅ๐‘ฆ๐‘งโ‰ โˆ’1. We fall short of finding a necessary and sufficient condition in the case ๐‘˜=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 ๐‘˜irreducible representations of the braid group, where ๐‘˜โ‰ฅ2. Therefore, our paper is concerned with the tensor product of the following irreducible representations:๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎€ธ,๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—ฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’2โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎€ธ,โ‹ฎฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—ฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—ฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’2โŠ—โ„‚๐‘›โˆ’2โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’2๎€ธ.(1.2)

Our main result is that, for ๐‘›โ‰ฅ4, the above representations are irreducible if (โˆ’1)๐‘™๐‘ค๐‘™โ‰ (โˆ’1)๐‘ก๐‘ค๐‘กยฑ1, for ๐‘™,๐‘กโˆˆ{1,โ€ฆ,๐‘˜}, where ๐‘ค๐‘™ and ๐‘ค๐‘ก are positive words of lengths ๐‘™ and ๐‘ก,respectively, and do not have any ๐‘ฅ๐‘–in common.

2. Definitions

The braid group on ๐‘› strings, ๐ต๐‘›, is defined as an abstract group with ๐‘›โˆ’1 generators ๐œŽ๐‘– (๐‘–=1,2,โ€ฆ,๐‘›โˆ’1) and relations:(i)๐œŽ๐‘–๐œŽ๐‘–+1๐œŽ๐‘–=๐œŽ๐‘–+1๐œŽ๐‘–๐œŽ๐‘–+1 for ๐‘–=1,2,โ€ฆ,๐‘›โˆ’2,(ii)๐œŽ๐‘–๐œŽ๐‘—=๐œŽ๐‘—๐œŽ๐‘–if|๐‘–โˆ’๐‘—|โ‰ฅ2.

The generators ๐œŽ1,๐œŽ2โ€ฆ,๐œŽ๐‘›โˆ’1 are called the standard generators. Let ๐‘ก be an indeterminate, and let โ„‚[๐‘กยฑ1] be a Laurent polynomial ring over the complex numbers. All modules are โ„‚-vector spaces, so ๐ต๐‘› modules and โ„‚[๐ต๐‘›] modules will mean the same. We define the following representations of ๐ต๐‘› by matrices over โ„‚[๐‘กยฑ1].

Definition 1. The reduced Burau representation ๐›ฝ๐‘›(๐‘ก)โˆถ๐ต๐‘›โ†’GL๐‘›โˆ’1(โ„‚[๐‘กยฑ1]) is given by ๐œŽ1(๐‘ก)=๐›ฝ๐‘›(๐‘ก)๎€ท๐œŽ1๎€ธ=โŽ›โŽœโŽœโŽœโŽœโŽโˆ’๐‘ก0โˆ’1100๐ผ๐‘›โˆ’3โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,๐œŽ๐‘–(๐‘ก)=๐›ฝ๐‘›(๐‘ก)๎€ท๐œŽ๐‘–๎€ธ=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ผ๐‘–โˆ’20001โˆ’๐‘ก00โˆ’๐‘ก00โˆ’11000๐ผ๐‘›โˆ’๐‘–โˆ’2โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,for๐‘–=2,โ€ฆ,๐‘›โˆ’2,๐œŽ๐‘›โˆ’1(๐‘ก)=๐›ฝ๐‘›(๐‘ก)๎€ท๐œŽ๐‘›โˆ’1๎€ธ=โŽ›โŽœโŽœโŽœโŽœโŽ๐ผ๐‘›โˆ’3001โˆ’๐‘ก0โˆ’๐‘กโŽžโŽŸโŽŸโŽŸโŽŸโŽ .(2.1)

Definition 2. ๐œ๐‘›=๐œŽ1๐œŽ2โ‹ฏ๐œŽ๐‘›โˆ’1and ๐œ๐‘›(๐‘ก)=๐œŽ1(๐‘ก)๐œŽ2(๐‘ก)โ‹ฏ๐œŽ๐‘›โˆ’1(๐‘ก).

Direct calculations show that๐œ๐‘›(๐‘ก)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโˆ’๐‘ก๐‘ก2โˆ’๐‘ก3โ‹ฏ(โˆ’๐‘ก)๐‘›โˆ’2(โˆ’๐‘ก)๐‘›โˆ’1โˆ’100โ‹ฏ000โˆ’10โ‹ฏ0000โˆ’1โ‹ฏ00000โ‹ฏโˆ’10โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(2.2)

We identify โ„‚๐‘›โˆ’1 with (๐‘›โˆ’1)ร—1 column vectors, we let ๐‘’1,๐‘’2,โ€ฆ,๐‘’๐‘›โˆ’1 denote the standard basis for โ„‚๐‘›โˆ’1, and we consider matrices to act by left multiplication on column vectors.

Definition 3. If ๐‘Ÿ=๐‘Ž1๐‘’1+๐‘Ž2๐‘’2+โ‹ฏ+๐‘Ž๐‘›โˆ’1๐‘’๐‘›โˆ’1โˆˆโ„‚๐‘›โˆ’1, the support of ๐‘Ÿ, denoted by supp(๐‘Ÿ), is the set {๐‘’๐‘–โˆฃ๐‘Ž๐‘–โ‰ 0}. If ๐‘ =โˆ‘๐‘Ž๐‘–๐‘—(๐‘’๐‘–โŠ—๐‘’๐‘—)โˆˆโ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1, the support of ๐‘ , also denoted by supp(๐‘ ), is the set {๐‘’๐‘–โŠ—๐‘’๐‘—โˆฃ๐‘Ž๐‘–๐‘—โ‰ 0}, and ๐‘Ž๐‘–๐‘— is called the coefficient of๐‘’๐‘–โŠ—๐‘’๐‘— in ๐‘ .

Definition 4. For ๐‘–=1,2,โ€ฆ,๐‘›โˆ’1,๐‘ฃ๐‘–(๐‘ก)=๐‘’๐‘–โˆ’๐œŽ๐‘–(๐‘ก)(๐‘’๐‘–)=(๐ผโˆ’๐œŽ๐‘–(๐‘ก))(๐‘’๐‘–).

3. Preliminaries

Lemma 3.1. (a) ๐œŽ๐‘–(๐‘ก)(๐‘ฃ๐‘–โˆ’1(๐‘ก))=๐‘ฃ๐‘–โˆ’1(๐‘ก)โˆ’๐‘ฃ๐‘–(๐‘ก) for 2โ‰ค๐‘–โ‰ค๐‘›โˆ’1,
๐œŽ๐‘–(๐‘ก)(๐‘ฃ๐‘–(๐‘ก))=โˆ’๐‘ก๐‘ฃ๐‘–(๐‘ก) for 1โ‰ค๐‘–โ‰ค๐‘›โˆ’1,
๐œŽ๐‘–(๐‘ก)(๐‘ฃ๐‘–+1(๐‘ก))=โˆ’๐‘ก๐‘ฃ๐‘–(๐‘ก)+๐‘ฃ๐‘–+1(๐‘ก) for 1โ‰ค๐‘–โ‰ค๐‘›โˆ’2,
๐œŽ๐‘–(๐‘ก)(๐‘ฃ๐‘—(๐‘ก))=๐‘ฃ๐‘—(๐‘ก) for 1โ‰ค๐‘–,โ€‰โ€‰๐‘—โ‰ค๐‘›โˆ’1,โ€‰โ€‰|๐‘–โˆ’๐‘—|โ‰ฅ2,
(b) ๐œ๐‘›(๐‘ฃ๐‘–(๐‘ก))=โˆ’๐‘ฃ๐‘–+1(๐‘ก) for 1โ‰ค๐‘–โ‰ค๐‘›โˆ’2.

Note that the above lemma remains true for any specialization ๐‘กโ†’๐‘ฆ, where ๐‘ฆโˆˆโ„‚โˆ—.

Notation 1. Fix nonzero complex numbers ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘˜. Let ๐œŽโˆˆ๐ต๐‘›and ๐‘ข=๐‘ข1โŠ—โ‹ฏโŠ—๐‘ข๐‘˜โˆˆโ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1, then we write ๐œŽ(๐‘ข1โŠ—โ‹ฏโŠ—๐‘ข๐‘˜)=๐œŽ(๐‘ฅ1)(๐‘ข1)โŠ—โ‹ฏโŠ—๐œŽ(๐‘ฅ๐‘˜)(๐‘ข๐‘˜).

Notation 2. Given ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘˜โˆˆโ„‚โˆ—, by a positive word of length ๐‘™, we mean a word that is written as a product of ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘˜, where the number of ๐‘ฅ๐‘–s involved is ๐‘™, and their exponents are ones. Each ๐‘ฅ๐‘–in ๐‘ค๐‘™appears exactly once. We denote the word by ๐‘ค๐‘™. As an example, we write ๐‘ค3=๐‘ฅ1๐‘ฅ3๐‘ฅ4to stand for a word of length 3.

Lemma 3.2. ๐œŽ๐‘›๐‘–โŽ›โŽœโŽœโŽ๐‘’๐‘–โŠ—๐‘’๐‘–โŠ—โ‹ฏโŠ—๐‘’๐‘–๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesโŽžโŽŸโŽŸโŽ =โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎‚ต๐‘’๐‘–โˆ’๎‚ต1+๐‘ฅ๐‘›11+๐‘ฅ1๎‚ถ๐‘ฃ๐‘–๎€ท๐‘ฅ1๎€ธ๎‚ถโŠ—โ‹ฏโŠ—๎‚ต๐‘’๐‘–โˆ’๎‚ต1+๐‘ฅ๐‘›๐‘˜1+๐‘ฅ๐‘˜๎‚ถ๐‘ฃ๐‘–๎€ท๐‘ฅ๐‘–๐‘˜๎€ธ๎‚ถif๐‘›isodd,๎‚ต๐‘’๐‘–โˆ’๎‚ต1โˆ’๐‘ฅ๐‘›11+๐‘ฅ1๎‚ถ๐‘ฃ๐‘–๎€ท๐‘ฅ1๎€ธ๎‚ถโŠ—โ‹ฏโŠ—๎‚ต๐‘’๐‘–โˆ’๎‚ต1โˆ’๐‘ฅ๐‘›๐‘˜1+๐‘ฅ๐‘˜๎‚ถ๐‘ฃ๐‘–๎€ท๐‘ฅ๐‘˜๎€ธ๎‚ถif๐‘›iseven.(3.1)

Lemma 3.3. Let ๐›ฝ๐‘›(๐‘ฆ)โˆถ๐ต๐‘›โ†’GL(โ„‚๐‘›โˆ’1) be a specialization of the Burau representation making โ„‚๐‘›โˆ’1 into a ๐ต๐‘›-module, where ๐‘›โ‰ฅ3, then (a)let ๐ด be the kernel of the homomorphism โ„‚[๐ต๐‘›]โ†’โ„‚ induced by ๐œŽ๐‘–โ†’1 (the augmentation ideal), then ๐ดโ„‚๐‘›โˆ’1 is equal to the โ„‚-vector space spanned by ๐‘ฃ1(๐‘ฆ),โ€ฆ,๐‘ฃ๐‘›โˆ’1(๐‘ฆ),(b)if ๐‘€ is a nonzero ๐ต๐‘›-submodule of โ„‚๐‘›โˆ’1, then ๐ดโ„‚๐‘›โˆ’1โŠ†๐‘€. Hence, ๐ดโ„‚๐‘›โˆ’1 is the unique minimal nonzero ๐ต๐‘›-module of โ„‚๐‘›โˆ’1,(c)if ๐‘ฆ is not a root of ๐‘(๐‘ก)=๐‘ก๐‘›โˆ’1+๐‘ก๐‘›โˆ’2+โ‹ฏ+๐‘ก+1, then ๐ดโ„‚๐‘›โˆ’1=โ„‚๐‘›โˆ’1, and ๐›ฝ๐‘›(๐‘ฆ) is irreducible. If ๐‘ฆ is a root of ๐‘(๐‘ก), then dim๐ถ(๐ดโ„‚๐‘›โˆ’1)=๐‘›โˆ’2.

Hence, ๐ดโ„‚๐‘›โˆ’1 is its unique minimal nonzero ๐ต๐‘›-submodule. Of course, ๐ดโ„‚๐‘›โˆ’1=โ„‚๐‘›โˆ’1 when ๐›ฝ๐‘›(๐‘ฆ) is irreducible, but when ๐›ฝ๐‘›(๐‘ฆ) is reducible, ๐ดโ„‚๐‘›โˆ’1 is the subrepresentation ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฆ).

The main technical result is Proposition 4.1, which says that if (โˆ’1)๐‘™๐‘ค๐‘™โ‰ (โˆ’1)๐‘ก๐‘ค๐‘กยฑ1,for ๐‘™,๐‘กโˆˆ{1,โ€ฆ,๐‘˜},where ๐‘ค๐‘™and ๐‘ค๐‘กare positive words that do not have any ๐‘ฅ๐‘–in common, then ๐ดโ„‚๐‘›โˆ’1โŠ—๐ดโ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—๐ดโ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesis the unique minimal nonzero ๐ต๐‘›-module of โ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-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 ๐‘˜ Irreducible Representations (๐‘˜โ‰ฅ2)

We obtain a result concerning the irreducibility of the tensor product of ๐‘˜irreducible representations of the braid group, ๐ต๐‘›, where ๐‘˜โ‰ฅ2. We state our proposition and give an outline of a proof that goes along the same lines as in the cases ๐‘˜=2and ๐‘˜=3. However, a more general adequate proof is required here, which could be easily verified in the cases ๐‘˜=2and ๐‘˜=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 ๐‘€ be a nonzero ๐ต๐‘› submodule of โ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times under the action of ๐›ฝ๐‘›(๐‘ฅ1)โŠ—๐›ฝ๐‘›(๐‘ฅ2)โŠ—โ‹ฏโŠ—๐›ฝ๐‘›(๐‘ฅ๐‘˜)โˆถ๐ต๐‘›โ†’GL(โ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1), where ๐‘›โ‰ฅ3. Suppose that for ๐‘™,๐‘กโˆˆ{1,โ€ฆ,๐‘˜} one has (โˆ’1)๐‘™๐‘ค๐‘™โ‰ (โˆ’1)๐‘ก๐‘ค๐‘กยฑ1๎€ท๐‘ค๐‘™and๐‘ค๐‘กdonothaveany๐‘ฅ๐‘–incommon๎€ธ,(4.1) then, ๐‘€ contains all ๐‘ฃ๐‘ก1(๐‘ฅ1)โŠ—๐‘ฃ๐‘ก2(๐‘ฅ2)โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ก๐‘˜(๐‘ฅ๐‘˜) for 1โ‰ค๐‘ก1,๐‘ก2,โ€ฆ,๐‘ก๐‘˜โ‰ค๐‘›โˆ’1, and ๐‘€ contains ๐ดโ„‚๐‘›โˆ’1โŠ—๐ดโ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—๐ดโ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times, where the action of ๐ต๐‘› on the ๐‘–th factor is induced by ๐›ฝ๐‘›(๐‘ฅ๐‘–).

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 ๐‘˜โ‰ฅ2.

Claim 1. There exists ๐‘šโˆˆ๐‘€ such that ๐‘’1โŠ—๐‘’1โŠ—โ‹ฏโŠ—๐‘’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesor ๐‘’2โŠ—๐‘’2โŠ—โ‹ฏโŠ—๐‘’2๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesโˆˆsupp(๐‘š).

Proof of Claim 1. When ๐‘˜=2, we set ๐ฝ=๎€ฝ๐‘โˆฃ๐‘’๐‘โŠ—๐‘’๐‘žor๐‘’๐‘žโŠ—๐‘’๐‘โˆˆsupp(๐‘š)forsome๐‘šโˆˆ๐‘€andsome๐‘ž๎€พ.(4.2)
First, we show that 1โˆˆ๐ฝ. Second, we let ๐‘=๐œŽ3โ‹ฏ๐œŽ๐‘˜, where ๐‘˜=min{๐‘โˆฃ๐‘’๐‘โˆˆsupp(๐‘ฃ)โˆชsupp(๐‘ค)}. Using some recursive argument, we show that there is an element ๐‘šor ๐‘(๐‘š)of the form ๐‘š=๐‘Ž๎€ท๐‘’1โŠ—๐‘’2๎€ธ+๐‘๎€ท๐‘’2โŠ—๐‘’1๎€ธ+๐‘Š,(4.3) where ๐‘’1โŠ—๐‘’1,๐‘’1โŠ—๐‘’2,๐‘’2โŠ—๐‘’1โˆ‰supp(๐‘Š), and at least one of ๐‘Ž,๐‘ is nonzero. If ๐‘’2โŠ—๐‘’2โˆˆsupp(๐‘Š), we are done. If ๐‘’2โŠ—๐‘’2โˆ‰supp(๐‘Š), then โˆ’๐‘Ž๐‘งโˆ’by=coe๏ฌƒcientof๐‘’1โŠ—๐‘’1in๐œŽ2(๐‘š),๐‘Ž๎€ท๐‘ง2โˆ’๐‘ง๎€ธ+๐‘๎€ท๐‘ฆ2โˆ’๐‘ฆ๎€ธ=coe๏ฌƒcientof๐‘’1โŠ—๐‘’1in๎€ท๐œŽ2๎€ธ2(๐‘š).(4.4) The determinant detโŽ›โŽœโŽœโŽโˆ’๐‘งโˆ’๐‘ฆ๐‘ง2โˆ’๐‘ง๐‘ฆ2โˆ’๐‘ฆโŽžโŽŸโŽŸโŽ =๐‘ฆ๐‘ง(๐‘งโˆ’๐‘ฆ),(4.5) is nonzero, since ๐‘ฆโ‰ ๐‘ง. Then one of โˆ’๐‘Ž๐‘งโˆ’by,๐‘Ž(๐‘ง2โˆ’๐‘ง)+๐‘(๐‘ฆ2โˆ’๐‘ฆ) is nonzero, and one of ๐œŽ2(๐‘š), (๐œŽ2)2(๐‘š) has ๐‘’1โŠ—๐‘’1 in its support. For more details, see [3].

As for the general case ๐‘˜โ‰ฅ2, we follow the same argument as above. The 2ร—2above is generalized to a (2๐‘˜โˆ’2)ร—(2๐‘˜โˆ’2) matrix ๐ด๐‘˜ whose rows ๐‘…1,๐‘…2,โ€ฆ,๐‘…2๐‘˜โˆ’2 are given by๐‘…1=๎€ฝ(โˆ’1)๐‘™๐‘ค๐‘™,1โ‰ค๐‘™โ‰ค๐‘˜โˆ’1๎€พ,๐‘…๐‘–=๎€ฝ๎€ฝ๐‘“๐‘–๎€ท๐‘ค๐‘™๎€ธ,1โ‰ค๐‘™โ‰ค๐‘˜โˆ’1,๐‘ค๐‘™arethesamepositivewordsin๐‘…1๎€พ,(4.6) for ๐‘–=2,โ€ฆ,2๐‘˜โˆ’2.

Here, ๐‘“๐‘–(๐‘ฅ๐‘™)=๐›ผ๐‘–(๐‘ฅ๐‘™) for ๐‘–=1,โ€ฆ,2๐‘˜โˆ’2,and ๐‘ค๐‘™s are all positive words of length ๐‘™. Using the hypothesis, the determinant of the matrix ๐ด๐‘˜ is nonzero, and this gives that one of the coefficients in ๐œŽ๐‘–2(๐‘š) has ๐‘’1โŠ—๐‘’1โŠ—โ‹ฏโŠ—๐‘’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesโˆˆsupp(๐‘š) for ๐‘–=1,2,โ€ฆ,2๐‘˜โˆ’2.

Claim 2. Suppose that ๐‘’๐‘–โŠ—๐‘’๐‘–โŠ—โ‹ฏโŠ—๐‘’๐‘–๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesโˆˆsupp(๐‘š) for some ๐‘šโˆˆ๐‘€, then ๐‘ฃ๐‘–โŠ—๐‘ฃ๐‘–โŠ—โ‹ฏโŠ—๐‘ฃ๐‘–๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesโˆˆ๐‘€.

Proof of Claim 2. ๎€ท๐œŽ๐‘–โˆ’1๎€ธ๐‘˜โˆ’1โˆ๐‘ก=1๎€ท๐œŽ๐‘–โˆ’(โˆ’1)๐‘ก๐‘ค๐‘ก๎€ธ๎€ท๐‘’๐‘–โŠ—๐‘’๐‘–โŠ—โ‹ฏโŠ—๐‘’๐‘–๎€ธ๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times=๐‘˜โˆ๐‘ก=2๎€ท๐‘ค๐‘กโˆ’(โˆ’1)๐‘ก๎€ธโˆ๎€ทโˆ’๐‘ค1๎€ธ2๐‘˜โˆ’1โˆ’1(๐‘ฃ๐‘–โŠ—โ‹ฏโŠ—๐‘ฃ๐‘–)๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times,๎€ท๐œŽ๐‘–โˆ’1๎€ธ๐‘˜โˆ’1๎‘๐‘ก=1๎€ท๐œŽ๐‘–โˆ’(โˆ’1)๐‘ก๐‘ค๐‘ก๎€ธ๎€ท๐‘’๐‘–1โŠ—๐‘’๐‘–2โŠ—โ‹ฏโŠ—๐‘’๐‘–๐‘˜๎€ธ=0if๎€ท๐‘–1,โ€ฆ,๐‘–๐‘˜๎€ธโ‰ (๐‘–,โ€ฆ,๐‘–).(4.7) Here, ๐‘ค๐‘กrepresents all words of length ๐‘ก. By our hypothesis, the proof is done.

Claim 3. There exists at least one (๐‘ 1,๐‘ 2,โ€ฆ,๐‘ ๐‘˜) such that ๐‘ฃ๐‘ 1โŠ—๐‘ฃ๐‘ 2โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ ๐‘˜โˆˆ๐‘€, where ๐‘ 1,๐‘ 2,โ€ฆ,๐‘ ๐‘˜โˆˆ{1,2} and (๐‘ 1,๐‘ 2,โ€ฆ,๐‘ ๐‘˜) is neither (1,1,โ€ฆ,1) nor (2,2,โ€ฆ,2).

Proof of Claim 3. Knowing that ๐‘ฃ1โŠ—๐‘ฃ1โŠ—โ‹ฏโŠ—๐‘ฃ1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-timesโˆˆ๐‘€, then ๐œŽ2(๐‘ฃ1โŠ—๐‘ฃ1โŠ—โ‹ฏโŠ—๐‘ฃ1)โˆˆ๐‘€. It follows that ๎€ท๐‘ฃ1โˆ’๐‘ฃ2๎€ธโŠ—๎€ท๐‘ฃ1โˆ’๐‘ฃ2๎€ธโŠ—โ‹ฏโŠ—๎€ท๐‘ฃ1โˆ’๐‘ฃ2๎€ธโˆˆ๐‘€.(4.8)
The proof is completed by applying ๐œŽ1 repeatedly to the expression above.

Claim 4. All the tensors ๐‘ฃ๐‘ 1โŠ—๐‘ฃ๐‘ 2โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ ๐‘˜โˆˆ๐‘€ for all ๐‘ 1,๐‘ 2,โ€ฆ,๐‘ ๐‘˜โˆˆ{1,2}.

Proof of Claim 4. By applying ๐œŽ2 to the tensors obtained from Claim 3 and then applying ๐œŽ1 repeatedly, the proof is done.

Claim 5. For 3โ‰ค๐‘–โ‰ค๐‘›โˆ’2, all tensors of the form ๐‘ฃ๐‘–โŠ—๐‘ฃ1โŠ—โ‹ฏโŠ—๐‘ฃ1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-1times and ๐‘ฃ๐‘–โŠ—๐‘ฃ2โŠ—โ‹ฏโŠ—๐‘ฃ2๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-1times and those obtained by permuting the indices are in ๐‘€.

Proof of Claim 5. Knowing that ๐œŽ๐‘–+1๎€ท๐‘ฃ๐‘–๎€ท๐‘ฅ1๎€ธโŠ—๐‘ฃ1๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐‘ฃ1๎€ท๐‘ฅ๐‘˜๎€ธ๎€ธ=๎€ท๐‘ฃ๐‘–๎€ท๐‘ฅ1๎€ธโˆ’๐‘ฃ๐‘–+1๎€ท๐‘ฅ1๎€ธ๎€ธโŠ—๐‘ฃ1๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐‘ฃ1๎€ท๐‘ฅ๐‘˜๎€ธ,๐œŽ๐‘–+1๎€ท๐‘ฃ๐‘–๎€ท๐‘ฅ1๎€ธโŠ—๐‘ฃ2๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐‘ฃ2๎€ท๐‘ฅ๐‘˜๎€ธ๎€ธ=๎€ท๐‘ฃ๐‘–(๐‘ฅ)โˆ’๐‘ฃ๐‘–+1(๐‘ฅ)๎€ธโŠ—๐‘ฃ2๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐‘ฃ2๎€ท๐‘ฅ๐‘˜๎€ธ,(4.9) we apply induction on ๐‘–, and the proof is finished.

Claim 6. All tensors ๐‘ฃ๐‘–โŠ—๐‘ฃ๐‘—โŠ—๐‘ฃ1โŠ—โ‹ฏโŠ—๐‘ฃ1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-2times and those obtained by permuting the indices are in ๐‘€ for ๐‘–,๐‘—โ‰ฅ3.

Proof of Claim 6. This is done by induction on ๐‘— and using Claim 5.

Claim 7. All tensors of the form ๐‘ฃ๐‘–โŠ—๐‘ฃ๐‘—โŠ—๐‘ฃโ„ŽโŠ—๐‘ฃ1โŠ—๐‘ฃ1โŠ—โ‹ฏโŠ—๐‘ฃ1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-3times and those obtained by permuting the indices are in ๐‘€.

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

Claim 8. All tensors ๐‘ฃ1โŠ—๐‘ฃ๐‘ 1โŠ—๐‘ฃ๐‘ 2โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ ๐‘˜โˆ’1 are in ๐‘€ for all 1โ‰ค๐‘ 1,๐‘ 2,โ€ฆ,๐‘ ๐‘˜โˆ’1โ‰ค๐‘›โˆ’1.

Proof of Claim 8. This is done by induction on ๐‘ ๐‘˜โˆ’1 and using the previous claims.

Claim 9. ๐‘ฃ๐‘ 1โŠ—๐‘ฃ๐‘ 2โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ ๐‘˜โˆˆ๐‘€ for all 1โ‰ค๐‘ 1,๐‘ 2,โ€ฆ,๐‘ ๐‘˜โˆ’1โ‰ค๐‘›โˆ’1.

Proof of Claim 9. By Claim 8, we have that all the tensors ๐‘ฃ1โŠ—๐‘ฃ๐‘ 1โŠ—๐‘ฃ๐‘ 2โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ ๐‘˜โˆ’1are in ๐‘€, Consider then ๐œ๐‘›๎€ท๐‘ฃ1โŠ—๐‘ฃ๐‘ 1โŠ—๐‘ฃ๐‘ 2โŠ—โ‹ฏโŠ—๐‘ฃ๐‘ ๐‘˜โˆ’1๎€ธโˆˆ๐‘€.(4.10) Knowing that ๐œ๐‘›(๐‘ฃ๐‘–(๐‘ก))=โˆ’๐‘ฃ๐‘–+1(๐‘ก), the proof is completed.

We now get our main theorem.

Theorem 4.2. Consider the tensor products of the irreducible representations ๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎€ธ,where๐‘๎€ท๐‘ฅ1๎€ธโ‰ 0,๐‘๎€ท๐‘ฅ2๎€ธโ‰ 0,โ€ฆ,๐‘๎€ท๐‘ฅ๐‘˜๎€ธโ‰ 0,๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—ฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’2โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎€ธ,where๐‘๎€ท๐‘ฅ1๎€ธโ‰ 0,๐‘๎€ท๐‘ฅ2๎€ธ=0,โ€ฆ,๐‘๎€ท๐‘ฅ๐‘˜๎€ธโ‰ 0,โ‹ฎฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—ฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—ฬ‚โ€Œ๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’2โŠ—โ„‚๐‘›โˆ’2โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’2๎€ธ,where๐‘๎€ท๐‘ฅ1๎€ธ=0,๐‘๎€ท๐‘ฅ2๎€ธ=0,โ€ฆ,๐‘๎€ท๐‘ฅ๐‘˜๎€ธ=0,(4.11) where ๐›ฝ๐‘›(๐‘ฅ)โˆถ๐ต๐‘›โ†’GL(โ„‚๐‘›โˆ’1) and ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฅ)โˆถ๐ต๐‘›โ†’GL(๐ดโ„‚๐‘›โˆ’1)=GL(โ„‚๐‘›โˆ’2) denote a specialization of the reduced Burau representation and the irreducible subrepresentation of Lemma 3.3(b), respectively. If, for ๐‘™,๐‘กโˆˆ{1,โ€ฆ,๐‘˜}, one has that (โˆ’1)๐‘™๐‘ค๐‘™โ‰ (โˆ’1)๐‘ก๐‘ค๐‘กยฑ1๎€ท๐‘ค๐‘™and๐‘ค๐‘กdonothaveany๐‘ฅ๐‘–incommon๎€ธ,(4.12) then the above representations are irreducible.

Proof. The proof is along the same lines as in the special cases ๐‘˜=2 and ๐‘˜=3. All of the above representations are subrepresentations of ๐›ฝ๐‘›๎€ท๐‘ฅ1๎€ธโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ2๎€ธโŠ—โ‹ฏโŠ—๐›ฝ๐‘›๎€ท๐‘ฅ๐‘˜๎€ธโˆถ๐ต๐‘›โŸถGL๎€ทโ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎€ธ.(4.13)
By Proposition 4.1, ๐ดโ„‚๐‘›โˆ’1โŠ—๐ดโ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—๐ดโ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times is the unique minimal nonzero ๐ต๐‘› module of โ„‚๐‘›โˆ’1โŠ—โ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—โ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times. In particular, it is an irreducible ๐ต๐‘›-module. By Lemma 3.3, the first factor ๐ดโ„‚๐‘›โˆ’1 corresponds to one of the representations ๐›ฝ๐‘›(๐‘ฅ1) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฅ1), the second factor ๐ดโ„‚๐‘›โˆ’1 corresponds to one of the representations ๐›ฝ๐‘›(๐‘ฅ2) or ฬ‚โ€Œ๐›ฝ๐‘›(๐‘ฅ2), and so on according to whether or not ๐‘ฅ1, ๐‘ฅ2,โ€ฆ๐‘ฅ๐‘˜ are roots of ๐‘(๐‘ก). Hence, all the above representations can be identified with the ๐ต๐‘›-module ๐ดโ„‚๐‘›โˆ’1โŠ—๐ดโ„‚๐‘›โˆ’1โŠ—โ‹ฏโŠ—๐ดโ„‚๐‘›โˆ’1๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘˜-times, so they are irreducible.

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