International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 104263 | https://doi.org/10.5402/2011/104263

Mohammad N. Abdulrahim, Wiaam M. Zeid, "Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups", International Scholarly Research Notices, vol. 2011, Article ID 104263, 9 pages, 2011. https://doi.org/10.5402/2011/104263

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

Academic Editor: A. KiliƧman
Received17 Aug 2011
Accepted05 Sep 2011
Published30 Oct 2011

Abstract

The reduced Burau representation is a one-parameter representation of šµš‘›, the braid group on š‘› strings. Specializing the parameter to nonzero complex number š‘„ gives a representation š›½š‘›(š‘„): šµš‘›ā†’šŗšæ(ā„‚š‘›āˆ’1), which is either irreducible or has an irreducible composition factor Ģ‚ā€Œš›½š‘›(š‘„): šµš‘›ā†’šŗšæ(ā„‚š‘›āˆ’2). In our paper, we let š‘˜ā‰„2, and we determine a sufficient condition for the irreducibility of the tensor product of š‘˜ irreducible Burau representations. This is a generalization of our previous work concerning the cases š‘˜=2 and š‘˜=3.

1. Introduction

Let šµš‘› be the braid group on š‘› strings. We consider the linear representation of šµš‘› called the Burau representation [1], which has a composition factor, the reduced Burau representationš›½š‘›(š‘”)āˆ¶šµš‘›āŸ¶GLš‘›āˆ’1ī€·ā„‚ī€ŗš‘”Ā±1ī€»ī€ø,(1.1) where š‘” is an indeterminate. Specializing š‘”ā†’š‘„, where š‘„āˆˆā„‚āˆ—, defines a representation š›½š‘›(š‘„)āˆ¶šµš‘›ā†’GLš‘›āˆ’1(ā„‚) which is either irreducible or has an irreducible subrepresentation Ģ‚ā€Œš›½š‘›(š‘„) of degree š‘›āˆ’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 š‘˜irreducible representations of the braid group, šµš‘›, irreducible? The question was answered in the cases š‘˜=2and š‘˜=3. We prove that the tensor product of an irreducible š›½š‘›(š‘¦) or Ģ‚ā€Œš›½š‘›(š‘¦) with an irreducible š›½š‘›(š‘§) or Ģ‚ā€Œš›½š‘›(š‘§) is irreducible if and only if š‘¦ā‰ š‘§Ā±1. For more details, see [3]. We also consider the case š‘˜=3and find a sufficient condition that guarantees the irreducibility of three irreducible representations of šµš‘›. In other words, we show that the tensor product of an irreducible š›½š‘›(š‘„) or Ģ‚ā€Œš›½š‘›(š‘„) with an irreducible š›½š‘›(š‘¦) or Ģ‚ā€Œš›½š‘›(š‘¦) with an irreducible š›½š‘›(š‘§) or Ģ‚ā€Œš›½š‘›(š‘§) is irreducible if š‘„ā‰ š‘¦Ā±1,ā€‰ā€‰š‘„ā‰ š‘§Ā±1,ā€‰ā€‰š‘¦ā‰ š‘§Ā±1, š‘„+š‘¦š‘§ā‰ 0,ā€‰ā€‰š‘¦+š‘„š‘§ā‰ 0,ā€‰ā€‰š‘§+š‘„š‘¦ā‰ 0,ā€‰ā€‰and š‘„š‘¦š‘§ā‰ āˆ’1. We fall short of finding a necessary and sufficient condition in the case š‘˜=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 š‘˜irreducible representations of the braid group, where š‘˜ā‰„2. Therefore, our paper is concerned with the tensor product of the following irreducible representations:š›½š‘›ī€·š‘„1ī€øāŠ—š›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī€ø,š›½š‘›ī€·š‘„1ī€øāŠ—Ģ‚ā€Œš›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’2āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī€ø,ā‹®Ģ‚ā€Œš›½š‘›ī€·š‘„1ī€øāŠ—Ģ‚ā€Œš›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—Ģ‚ā€Œš›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’2āŠ—ā„‚š‘›āˆ’2āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’2ī€ø.(1.2)

Our main result is that, for š‘›ā‰„4, the above representations are irreducible if (āˆ’1)š‘™š‘¤š‘™ā‰ (āˆ’1)š‘”š‘¤š‘”Ā±1, for š‘™,š‘”āˆˆ{1,ā€¦,š‘˜}, where š‘¤š‘™ and š‘¤š‘” are positive words of lengths š‘™ and š‘”,respectively, and do not have any š‘„š‘–in common.

2. Definitions

The braid group on š‘› strings, šµš‘›, is defined as an abstract group with š‘›āˆ’1 generators šœŽš‘– (š‘–=1,2,ā€¦,š‘›āˆ’1) and relations:(i)šœŽš‘–šœŽš‘–+1šœŽš‘–=šœŽš‘–+1šœŽš‘–šœŽš‘–+1 for š‘–=1,2,ā€¦,š‘›āˆ’2,(ii)šœŽš‘–šœŽš‘—=šœŽš‘—šœŽš‘–if|š‘–āˆ’š‘—|ā‰„2.

The generators šœŽ1,šœŽ2ā€¦,šœŽš‘›āˆ’1 are called the standard generators. Let š‘” be an indeterminate, and let ā„‚[š‘”Ā±1] be a Laurent polynomial ring over the complex numbers. All modules are ā„‚-vector spaces, so šµš‘› modules and ā„‚[šµš‘›] modules will mean the same. We define the following representations of šµš‘› by matrices over ā„‚[š‘”Ā±1].

Definition 1. The reduced Burau representation š›½š‘›(š‘”)āˆ¶šµš‘›ā†’GLš‘›āˆ’1(ā„‚[š‘”Ā±1]) is given by šœŽ1(š‘”)=š›½š‘›(š‘”)ī€·šœŽ1ī€ø=āŽ›āŽœāŽœāŽœāŽœāŽāˆ’š‘”0āˆ’1100š¼š‘›āˆ’3āŽžāŽŸāŽŸāŽŸāŽŸāŽ ,šœŽš‘–(š‘”)=š›½š‘›(š‘”)ī€·šœŽš‘–ī€ø=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽš¼š‘–āˆ’20001āˆ’š‘”00āˆ’š‘”00āˆ’11000š¼š‘›āˆ’š‘–āˆ’2āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,forš‘–=2,ā€¦,š‘›āˆ’2,šœŽš‘›āˆ’1(š‘”)=š›½š‘›(š‘”)ī€·šœŽš‘›āˆ’1ī€ø=āŽ›āŽœāŽœāŽœāŽœāŽš¼š‘›āˆ’3001āˆ’š‘”0āˆ’š‘”āŽžāŽŸāŽŸāŽŸāŽŸāŽ .(2.1)

Definition 2. šœš‘›=šœŽ1šœŽ2ā‹ÆšœŽš‘›āˆ’1and šœš‘›(š‘”)=šœŽ1(š‘”)šœŽ2(š‘”)ā‹ÆšœŽš‘›āˆ’1(š‘”).

Direct calculations show thatšœš‘›(š‘”)=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽāˆ’š‘”š‘”2āˆ’š‘”3ā‹Æ(āˆ’š‘”)š‘›āˆ’2(āˆ’š‘”)š‘›āˆ’1āˆ’100ā‹Æ000āˆ’10ā‹Æ0000āˆ’1ā‹Æ00000ā‹Æāˆ’10āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ .(2.2)

We identify ā„‚š‘›āˆ’1 with (š‘›āˆ’1)Ɨ1 column vectors, we let š‘’1,š‘’2,ā€¦,š‘’š‘›āˆ’1 denote the standard basis for ā„‚š‘›āˆ’1, and we consider matrices to act by left multiplication on column vectors.

Definition 3. If š‘Ÿ=š‘Ž1š‘’1+š‘Ž2š‘’2+ā‹Æ+š‘Žš‘›āˆ’1š‘’š‘›āˆ’1āˆˆā„‚š‘›āˆ’1, the support of š‘Ÿ, denoted by supp(š‘Ÿ), is the set {š‘’š‘–āˆ£š‘Žš‘–ā‰ 0}. If š‘ =āˆ‘š‘Žš‘–š‘—(š‘’š‘–āŠ—š‘’š‘—)āˆˆā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1, the support of š‘ , also denoted by supp(š‘ ), is the set {š‘’š‘–āŠ—š‘’š‘—āˆ£š‘Žš‘–š‘—ā‰ 0}, and š‘Žš‘–š‘— is called the coefficient ofš‘’š‘–āŠ—š‘’š‘— in š‘ .

Definition 4. For š‘–=1,2,ā€¦,š‘›āˆ’1,š‘£š‘–(š‘”)=š‘’š‘–āˆ’šœŽš‘–(š‘”)(š‘’š‘–)=(š¼āˆ’šœŽš‘–(š‘”))(š‘’š‘–).

3. Preliminaries

Lemma 3.1. (a) šœŽš‘–(š‘”)(š‘£š‘–āˆ’1(š‘”))=š‘£š‘–āˆ’1(š‘”)āˆ’š‘£š‘–(š‘”) for 2ā‰¤š‘–ā‰¤š‘›āˆ’1,
šœŽš‘–(š‘”)(š‘£š‘–(š‘”))=āˆ’š‘”š‘£š‘–(š‘”) for 1ā‰¤š‘–ā‰¤š‘›āˆ’1,
šœŽš‘–(š‘”)(š‘£š‘–+1(š‘”))=āˆ’š‘”š‘£š‘–(š‘”)+š‘£š‘–+1(š‘”) for 1ā‰¤š‘–ā‰¤š‘›āˆ’2,
šœŽš‘–(š‘”)(š‘£š‘—(š‘”))=š‘£š‘—(š‘”) for 1ā‰¤š‘–,ā€‰ā€‰š‘—ā‰¤š‘›āˆ’1,ā€‰ā€‰|š‘–āˆ’š‘—|ā‰„2,
(b) šœš‘›(š‘£š‘–(š‘”))=āˆ’š‘£š‘–+1(š‘”) for 1ā‰¤š‘–ā‰¤š‘›āˆ’2.

Note that the above lemma remains true for any specialization š‘”ā†’š‘¦, where š‘¦āˆˆā„‚āˆ—.

Notation 1. Fix nonzero complex numbers š‘„1,ā€¦,š‘„š‘˜. Let šœŽāˆˆšµš‘›and š‘¢=š‘¢1āŠ—ā‹ÆāŠ—š‘¢š‘˜āˆˆā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1, then we write šœŽ(š‘¢1āŠ—ā‹ÆāŠ—š‘¢š‘˜)=šœŽ(š‘„1)(š‘¢1)āŠ—ā‹ÆāŠ—šœŽ(š‘„š‘˜)(š‘¢š‘˜).

Notation 2. Given š‘„1,ā€¦,š‘„š‘˜āˆˆā„‚āˆ—, by a positive word of length š‘™, we mean a word that is written as a product of š‘„1,ā€¦,š‘„š‘˜, where the number of š‘„š‘–s involved is š‘™, and their exponents are ones. Each š‘„š‘–in š‘¤š‘™appears exactly once. We denote the word by š‘¤š‘™. As an example, we write š‘¤3=š‘„1š‘„3š‘„4to stand for a word of length 3.

Lemma 3.2. šœŽš‘›š‘–āŽ›āŽœāŽœāŽš‘’š‘–āŠ—š‘’š‘–āŠ—ā‹ÆāŠ—š‘’š‘–ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesāŽžāŽŸāŽŸāŽ =āŽ§āŽŖāŽŖāŽØāŽŖāŽŖāŽ©ī‚µš‘’š‘–āˆ’ī‚µ1+š‘„š‘›11+š‘„1ī‚¶š‘£š‘–ī€·š‘„1ī€øī‚¶āŠ—ā‹ÆāŠ—ī‚µš‘’š‘–āˆ’ī‚µ1+š‘„š‘›š‘˜1+š‘„š‘˜ī‚¶š‘£š‘–ī€·š‘„š‘–š‘˜ī€øī‚¶ifš‘›isodd,ī‚µš‘’š‘–āˆ’ī‚µ1āˆ’š‘„š‘›11+š‘„1ī‚¶š‘£š‘–ī€·š‘„1ī€øī‚¶āŠ—ā‹ÆāŠ—ī‚µš‘’š‘–āˆ’ī‚µ1āˆ’š‘„š‘›š‘˜1+š‘„š‘˜ī‚¶š‘£š‘–ī€·š‘„š‘˜ī€øī‚¶ifš‘›iseven.(3.1)

Lemma 3.3. Let š›½š‘›(š‘¦)āˆ¶šµš‘›ā†’GL(ā„‚š‘›āˆ’1) be a specialization of the Burau representation making ā„‚š‘›āˆ’1 into a šµš‘›-module, where š‘›ā‰„3, then (a)let š“ be the kernel of the homomorphism ā„‚[šµš‘›]ā†’ā„‚ induced by šœŽš‘–ā†’1 (the augmentation ideal), then š“ā„‚š‘›āˆ’1 is equal to the ā„‚-vector space spanned by š‘£1(š‘¦),ā€¦,š‘£š‘›āˆ’1(š‘¦),(b)if š‘€ is a nonzero šµš‘›-submodule of ā„‚š‘›āˆ’1, then š“ā„‚š‘›āˆ’1āŠ†š‘€. Hence, š“ā„‚š‘›āˆ’1 is the unique minimal nonzero šµš‘›-module of ā„‚š‘›āˆ’1,(c)if š‘¦ is not a root of š‘(š‘”)=š‘”š‘›āˆ’1+š‘”š‘›āˆ’2+ā‹Æ+š‘”+1, then š“ā„‚š‘›āˆ’1=ā„‚š‘›āˆ’1, and š›½š‘›(š‘¦) is irreducible. If š‘¦ is a root of š‘(š‘”), then dimš¶(š“ā„‚š‘›āˆ’1)=š‘›āˆ’2.

Hence, š“ā„‚š‘›āˆ’1 is its unique minimal nonzero šµš‘›-submodule. Of course, š“ā„‚š‘›āˆ’1=ā„‚š‘›āˆ’1 when š›½š‘›(š‘¦) is irreducible, but when š›½š‘›(š‘¦) is reducible, š“ā„‚š‘›āˆ’1 is the subrepresentation Ģ‚ā€Œš›½š‘›(š‘¦).

The main technical result is Proposition 4.1, which says that if (āˆ’1)š‘™š‘¤š‘™ā‰ (āˆ’1)š‘”š‘¤š‘”Ā±1,for š‘™,š‘”āˆˆ{1,ā€¦,š‘˜},where š‘¤š‘™and š‘¤š‘”are positive words that do not have any š‘„š‘–in common, then š“ā„‚š‘›āˆ’1āŠ—š“ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—š“ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesis the unique minimal nonzero šµš‘›-module of ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-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 š‘˜ Irreducible Representations (š‘˜ā‰„2)

We obtain a result concerning the irreducibility of the tensor product of š‘˜irreducible representations of the braid group, šµš‘›, where š‘˜ā‰„2. We state our proposition and give an outline of a proof that goes along the same lines as in the cases š‘˜=2and š‘˜=3. However, a more general adequate proof is required here, which could be easily verified in the cases š‘˜=2and š‘˜=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 š‘€ be a nonzero šµš‘› submodule of ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times under the action of š›½š‘›(š‘„1)āŠ—š›½š‘›(š‘„2)āŠ—ā‹ÆāŠ—š›½š‘›(š‘„š‘˜)āˆ¶šµš‘›ā†’GL(ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1), where š‘›ā‰„3. Suppose that for š‘™,š‘”āˆˆ{1,ā€¦,š‘˜} one has (āˆ’1)š‘™š‘¤š‘™ā‰ (āˆ’1)š‘”š‘¤š‘”Ā±1ī€·š‘¤š‘™andš‘¤š‘”donothaveanyš‘„š‘–incommonī€ø,(4.1) then, š‘€ contains all š‘£š‘”1(š‘„1)āŠ—š‘£š‘”2(š‘„2)āŠ—ā‹ÆāŠ—š‘£š‘”š‘˜(š‘„š‘˜) for 1ā‰¤š‘”1,š‘”2,ā€¦,š‘”š‘˜ā‰¤š‘›āˆ’1, and š‘€ contains š“ā„‚š‘›āˆ’1āŠ—š“ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—š“ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times, where the action of šµš‘› on the š‘–th factor is induced by š›½š‘›(š‘„š‘–).

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 š‘˜ā‰„2.

Claim 1. There exists š‘šāˆˆš‘€ such that š‘’1āŠ—š‘’1āŠ—ā‹ÆāŠ—š‘’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesor š‘’2āŠ—š‘’2āŠ—ā‹ÆāŠ—š‘’2ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesāˆˆsupp(š‘š).

Proof of Claim 1. When š‘˜=2, we set š½=ī€½š‘āˆ£š‘’š‘āŠ—š‘’š‘žorš‘’š‘žāŠ—š‘’š‘āˆˆsupp(š‘š)forsomeš‘šāˆˆš‘€andsomeš‘žī€¾.(4.2)
First, we show that 1āˆˆš½. Second, we let š‘=šœŽ3ā‹ÆšœŽš‘˜, where š‘˜=min{š‘āˆ£š‘’š‘āˆˆsupp(š‘£)āˆŖsupp(š‘¤)}. Using some recursive argument, we show that there is an element š‘šor š‘(š‘š)of the form š‘š=š‘Žī€·š‘’1āŠ—š‘’2ī€ø+š‘ī€·š‘’2āŠ—š‘’1ī€ø+š‘Š,(4.3) where š‘’1āŠ—š‘’1,š‘’1āŠ—š‘’2,š‘’2āŠ—š‘’1āˆ‰supp(š‘Š), and at least one of š‘Ž,š‘ is nonzero. If š‘’2āŠ—š‘’2āˆˆsupp(š‘Š), we are done. If š‘’2āŠ—š‘’2āˆ‰supp(š‘Š), then āˆ’š‘Žš‘§āˆ’by=coeļ¬ƒcientofš‘’1āŠ—š‘’1inšœŽ2(š‘š),š‘Žī€·š‘§2āˆ’š‘§ī€ø+š‘ī€·š‘¦2āˆ’š‘¦ī€ø=coeļ¬ƒcientofš‘’1āŠ—š‘’1inī€·šœŽ2ī€ø2(š‘š).(4.4) The determinant detāŽ›āŽœāŽœāŽāˆ’š‘§āˆ’š‘¦š‘§2āˆ’š‘§š‘¦2āˆ’š‘¦āŽžāŽŸāŽŸāŽ =š‘¦š‘§(š‘§āˆ’š‘¦),(4.5) is nonzero, since š‘¦ā‰ š‘§. Then one of āˆ’š‘Žš‘§āˆ’by,š‘Ž(š‘§2āˆ’š‘§)+š‘(š‘¦2āˆ’š‘¦) is nonzero, and one of šœŽ2(š‘š), (šœŽ2)2(š‘š) has š‘’1āŠ—š‘’1 in its support. For more details, see [3].

As for the general case š‘˜ā‰„2, we follow the same argument as above. The 2Ɨ2above is generalized to a (2š‘˜āˆ’2)Ɨ(2š‘˜āˆ’2) matrix š“š‘˜ whose rows š‘…1,š‘…2,ā€¦,š‘…2š‘˜āˆ’2 are given byš‘…1=ī€½(āˆ’1)š‘™š‘¤š‘™,1ā‰¤š‘™ā‰¤š‘˜āˆ’1ī€¾,š‘…š‘–=ī€½ī€½š‘“š‘–ī€·š‘¤š‘™ī€ø,1ā‰¤š‘™ā‰¤š‘˜āˆ’1,š‘¤š‘™arethesamepositivewordsinš‘…1ī€¾,(4.6) for š‘–=2,ā€¦,2š‘˜āˆ’2.

Here, š‘“š‘–(š‘„š‘™)=š›¼š‘–(š‘„š‘™) for š‘–=1,ā€¦,2š‘˜āˆ’2,and š‘¤š‘™s are all positive words of length š‘™. Using the hypothesis, the determinant of the matrix š“š‘˜ is nonzero, and this gives that one of the coefficients in šœŽš‘–2(š‘š) has š‘’1āŠ—š‘’1āŠ—ā‹ÆāŠ—š‘’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesāˆˆsupp(š‘š) for š‘–=1,2,ā€¦,2š‘˜āˆ’2.

Claim 2. Suppose that š‘’š‘–āŠ—š‘’š‘–āŠ—ā‹ÆāŠ—š‘’š‘–ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesāˆˆsupp(š‘š) for some š‘šāˆˆš‘€, then š‘£š‘–āŠ—š‘£š‘–āŠ—ā‹ÆāŠ—š‘£š‘–ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesāˆˆš‘€.

Proof of Claim 2. ī€·šœŽš‘–āˆ’1ī€øš‘˜āˆ’1āˆš‘”=1ī€·šœŽš‘–āˆ’(āˆ’1)š‘”š‘¤š‘”ī€øī€·š‘’š‘–āŠ—š‘’š‘–āŠ—ā‹ÆāŠ—š‘’š‘–ī€øī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times=š‘˜āˆš‘”=2ī€·š‘¤š‘”āˆ’(āˆ’1)š‘”ī€øāˆī€·āˆ’š‘¤1ī€ø2š‘˜āˆ’1āˆ’1(š‘£š‘–āŠ—ā‹ÆāŠ—š‘£š‘–)ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times,ī€·šœŽš‘–āˆ’1ī€øš‘˜āˆ’1ī‘š‘”=1ī€·šœŽš‘–āˆ’(āˆ’1)š‘”š‘¤š‘”ī€øī€·š‘’š‘–1āŠ—š‘’š‘–2āŠ—ā‹ÆāŠ—š‘’š‘–š‘˜ī€ø=0ifī€·š‘–1,ā€¦,š‘–š‘˜ī€øā‰ (š‘–,ā€¦,š‘–).(4.7) Here, š‘¤š‘”represents all words of length š‘”. By our hypothesis, the proof is done.

Claim 3. There exists at least one (š‘ 1,š‘ 2,ā€¦,š‘ š‘˜) such that š‘£š‘ 1āŠ—š‘£š‘ 2āŠ—ā‹ÆāŠ—š‘£š‘ š‘˜āˆˆš‘€, where š‘ 1,š‘ 2,ā€¦,š‘ š‘˜āˆˆ{1,2} and (š‘ 1,š‘ 2,ā€¦,š‘ š‘˜) is neither (1,1,ā€¦,1) nor (2,2,ā€¦,2).

Proof of Claim 3. Knowing that š‘£1āŠ—š‘£1āŠ—ā‹ÆāŠ—š‘£1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-timesāˆˆš‘€, then šœŽ2(š‘£1āŠ—š‘£1āŠ—ā‹ÆāŠ—š‘£1)āˆˆš‘€. It follows that ī€·š‘£1āˆ’š‘£2ī€øāŠ—ī€·š‘£1āˆ’š‘£2ī€øāŠ—ā‹ÆāŠ—ī€·š‘£1āˆ’š‘£2ī€øāˆˆš‘€.(4.8)
The proof is completed by applying šœŽ1 repeatedly to the expression above.

Claim 4. All the tensors š‘£š‘ 1āŠ—š‘£š‘ 2āŠ—ā‹ÆāŠ—š‘£š‘ š‘˜āˆˆš‘€ for all š‘ 1,š‘ 2,ā€¦,š‘ š‘˜āˆˆ{1,2}.

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

Claim 5. For 3ā‰¤š‘–ā‰¤š‘›āˆ’2, all tensors of the form š‘£š‘–āŠ—š‘£1āŠ—ā‹ÆāŠ—š‘£1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-1times and š‘£š‘–āŠ—š‘£2āŠ—ā‹ÆāŠ—š‘£2ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-1times and those obtained by permuting the indices are in š‘€.

Proof of Claim 5. Knowing that šœŽš‘–+1ī€·š‘£š‘–ī€·š‘„1ī€øāŠ—š‘£1ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š‘£1ī€·š‘„š‘˜ī€øī€ø=ī€·š‘£š‘–ī€·š‘„1ī€øāˆ’š‘£š‘–+1ī€·š‘„1ī€øī€øāŠ—š‘£1ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š‘£1ī€·š‘„š‘˜ī€ø,šœŽš‘–+1ī€·š‘£š‘–ī€·š‘„1ī€øāŠ—š‘£2ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š‘£2ī€·š‘„š‘˜ī€øī€ø=ī€·š‘£š‘–(š‘„)āˆ’š‘£š‘–+1(š‘„)ī€øāŠ—š‘£2ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š‘£2ī€·š‘„š‘˜ī€ø,(4.9) we apply induction on š‘–, and the proof is finished.

Claim 6. All tensors š‘£š‘–āŠ—š‘£š‘—āŠ—š‘£1āŠ—ā‹ÆāŠ—š‘£1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-2times and those obtained by permuting the indices are in š‘€ for š‘–,š‘—ā‰„3.

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

Claim 7. All tensors of the form š‘£š‘–āŠ—š‘£š‘—āŠ—š‘£ā„ŽāŠ—š‘£1āŠ—š‘£1āŠ—ā‹ÆāŠ—š‘£1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-3times and those obtained by permuting the indices are in š‘€.

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

Claim 8. All tensors š‘£1āŠ—š‘£š‘ 1āŠ—š‘£š‘ 2āŠ—ā‹ÆāŠ—š‘£š‘ š‘˜āˆ’1 are in š‘€ for all 1ā‰¤š‘ 1,š‘ 2,ā€¦,š‘ š‘˜āˆ’1ā‰¤š‘›āˆ’1.

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

Claim 9. š‘£š‘ 1āŠ—š‘£š‘ 2āŠ—ā‹ÆāŠ—š‘£š‘ š‘˜āˆˆš‘€ for all 1ā‰¤š‘ 1,š‘ 2,ā€¦,š‘ š‘˜āˆ’1ā‰¤š‘›āˆ’1.

Proof of Claim 9. By Claim 8, we have that all the tensors š‘£1āŠ—š‘£š‘ 1āŠ—š‘£š‘ 2āŠ—ā‹ÆāŠ—š‘£š‘ š‘˜āˆ’1are in š‘€, Consider then šœš‘›ī€·š‘£1āŠ—š‘£š‘ 1āŠ—š‘£š‘ 2āŠ—ā‹ÆāŠ—š‘£š‘ š‘˜āˆ’1ī€øāˆˆš‘€.(4.10) Knowing that šœš‘›(š‘£š‘–(š‘”))=āˆ’š‘£š‘–+1(š‘”), the proof is completed.

We now get our main theorem.

Theorem 4.2. Consider the tensor products of the irreducible representations š›½š‘›ī€·š‘„1ī€øāŠ—š›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī€ø,whereš‘ī€·š‘„1ī€øā‰ 0,š‘ī€·š‘„2ī€øā‰ 0,ā€¦,š‘ī€·š‘„š‘˜ī€øā‰ 0,š›½š‘›ī€·š‘„1ī€øāŠ—Ģ‚ā€Œš›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’2āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī€ø,whereš‘ī€·š‘„1ī€øā‰ 0,š‘ī€·š‘„2ī€ø=0,ā€¦,š‘ī€·š‘„š‘˜ī€øā‰ 0,ā‹®Ģ‚ā€Œš›½š‘›ī€·š‘„1ī€øāŠ—Ģ‚ā€Œš›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—Ģ‚ā€Œš›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’2āŠ—ā„‚š‘›āˆ’2āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’2ī€ø,whereš‘ī€·š‘„1ī€ø=0,š‘ī€·š‘„2ī€ø=0,ā€¦,š‘ī€·š‘„š‘˜ī€ø=0,(4.11) where š›½š‘›(š‘„)āˆ¶šµš‘›ā†’GL(ā„‚š‘›āˆ’1) and Ģ‚ā€Œš›½š‘›(š‘„)āˆ¶šµš‘›ā†’GL(š“ā„‚š‘›āˆ’1)=GL(ā„‚š‘›āˆ’2) denote a specialization of the reduced Burau representation and the irreducible subrepresentation of Lemma 3.3(b), respectively. If, for š‘™,š‘”āˆˆ{1,ā€¦,š‘˜}, one has that (āˆ’1)š‘™š‘¤š‘™ā‰ (āˆ’1)š‘”š‘¤š‘”Ā±1ī€·š‘¤š‘™andš‘¤š‘”donothaveanyš‘„š‘–incommonī€ø,(4.12) then the above representations are irreducible.

Proof. The proof is along the same lines as in the special cases š‘˜=2 and š‘˜=3. All of the above representations are subrepresentations of š›½š‘›ī€·š‘„1ī€øāŠ—š›½š‘›ī€·š‘„2ī€øāŠ—ā‹ÆāŠ—š›½š‘›ī€·š‘„š‘˜ī€øāˆ¶šµš‘›āŸ¶GLī€·ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī€ø.(4.13)
By Proposition 4.1, š“ā„‚š‘›āˆ’1āŠ—š“ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—š“ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times is the unique minimal nonzero šµš‘› module of ā„‚š‘›āˆ’1āŠ—ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times. In particular, it is an irreducible šµš‘›-module. By Lemma 3.3, the first factor š“ā„‚š‘›āˆ’1 corresponds to one of the representations š›½š‘›(š‘„1) or Ģ‚ā€Œš›½š‘›(š‘„1), the second factor š“ā„‚š‘›āˆ’1 corresponds to one of the representations š›½š‘›(š‘„2) or Ģ‚ā€Œš›½š‘›(š‘„2), and so on according to whether or not š‘„1, š‘„2,ā€¦š‘„š‘˜ are roots of š‘(š‘”). Hence, all the above representations can be identified with the šµš‘›-module š“ā„‚š‘›āˆ’1āŠ—š“ā„‚š‘›āˆ’1āŠ—ā‹ÆāŠ—š“ā„‚š‘›āˆ’1ī„æī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…ƒī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…€ī…Œš‘˜-times, so they are irreducible.

References

  1. J. S. Birman, Braids, Links, and Mapping Class Groups, vol. 8 of Annals of mathematics studies, Princeton University Press, Princeton, NJ, USA, 1975.
  2. E. Formanek, ā€œBraid group representations of low degree,ā€ Proceedings of the London Mathematical Society, vol. 73, no. 2, pp. 279ā€“322, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. M. N. Abdulrahim and E. Formanek, ā€œTensor products of specializations of the Burau representation,ā€ Journal of Pure and Applied Algebra, vol. 203, no. 1–3, pp. 104ā€“112, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. M. N. Abdulrahim, ā€œTensoring three irreducible linear representations of the braid group,ā€ Linear and Multilinear Algebra, vol. 57, no. 8, pp. 749ā€“758, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 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.


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