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

The reduced Burau representation is a one-parameter representation of Bn, 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 ̂ βn 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.


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 representation 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 β 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 k irreducible representations of the braid group, B n , irreducible?The . . .

1.2
Our main result is that, for n ≥ 4, the above representations are irreducible if −1 l w l / −1 t w 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 i in common.

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: The generators σ 1 , σ 2 . . ., 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 Direct calculations show that We identify C n−1 with n − 1 × 1 column vectors, we let e 1 , e 2 , . . ., e n−1 denote the standard basis for C n−1 , and we consider matrices to act by left multiplication on column vectors.
, 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 of e i ⊗ e j in s.

Preliminaries
Note that the above lemma remains true for any specialization t → y, where y Notation 2. Given x 1 , . . ., x k ∈ C * , by a positive word of length l, we mean a word that is written as a product of x 1 , . . ., x k , where the number of x i s involved is l, and their exponents are ones.Each x i in w l appears exactly once.We denote the word by w l .As an example, we write w 3 x 1 x 3 x 4 to stand for a word of length 3.
The main technical result is Proposition 4.1, which says that if −1 l w l / −1 t w t ±1 , for l, t ∈ {1, . . ., k}, where w l and w t are positive words that do not have any . 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.

Tensor Product of k Irreducible Representations k ≥ 2
We obtain a result concerning the irreducibility of the tensor product of k irreducible 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 2 and k 3.However, a more general adequate proof is required here, which could be easily verified in the cases k 2 and k 3 by 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.
under the action of where n ≥ 3. Suppose that for l, t ∈ {1, . . ., k} one has −1 l w l / −1 t w t ±1 w l and w t do not have any x i in common , 4.1 , 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. As for the general case k ≥ 2, we follow the same argument as above.The 2 Here, f i x l α i x l for i 1, . . ., 2 k − 2, and w l s 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 Proof of Claim 2.

4.7
Here, w t represents all words of length t.By our hypothesis, the proof is done.
Claim 3.There exists at least one The proof is completed by applying σ 1 repeatedly to the expression above.
Proof of Claim 4. By applying σ 2 to the tensors obtained from Claim 3 and then applying σ 1 repeatedly, the proof is done.
Claim 5.For 3 ≤ i ≤ n − 2, all tensors of the form and those obtained by permuting the indices are in M.
Proof of Claim 5. Knowing that 4.9 we apply induction on i, and the proof is finished.
and those obtained by permuting the indices are in Proof of Claim 6.This is done by induction on j and using Claim 5.
Claim 7. All tensors of the form 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.
Proof of Claim 8.This is done by induction on s k−1 and using the previous claims.
Proof of Claim 9.By Claim 8, we have that all the tensors Knowing that τ n v i t −v i 1 t , the proof is completed.
We now get our main theorem.
where p x 1 0, p x 2 0, . . ., p x k 0, 4.11 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 l w l / −1 t w t ±1 w l and w t do not have any x i in common , 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

4.13
By Proposition 4.1, . 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 x 1 or β n x 1 , the second factor AC n−1 corresponds to one of the representations β n x 2 or β n x 2 , and so on according to whether or not x 1 , x 2 , . . .x k are roots of p t .Hence, all the above representations can be identified with the B n -module , so they are irreducible.

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 v 1 y , . . ., v n−1 y ,

Claim 1 . 2 First, we show that 1 ∈ 3 where e 1 z 2 −
There exists m ∈ M such that e 1 ⊗ e 1 ⊗ • • • ⊗ e 1 k-times or e 2 ⊗ e 2 ⊗ • • • ⊗ e 2 k-times ∈ supp m .Proof of Claim 1.When k 2, we set J p | e p ⊗ e q or e q ⊗ e p ∈ supp m for some m ∈ M and some q .4.J.Second, we let p σ 3 • • • σ k , where k min{p | e p ∈ supp v ∪ supp w }.Using some recursive argument, we show that there is an element m or p m of the form m a e 1 ⊗ e 2 b e 2 ⊗ e 1 W, 4.⊗e 1 , e 1 ⊗e 2 , e 2 ⊗e 1 / ∈ supp W , and at least one of a, b is nonzero.If e 2 ⊗e 2 ∈ supp W , we are done.If e 2 ⊗ e 2 / ∈ supp W , then −az − by coefficient of e 1 ⊗ e 1 in σ 2 m , a z 2 − z b y 2 − y coefficient of e 1 ⊗ e 1 in σ 2 2 m .z y 2 − y yz z − y , 4.5 is nonzero, since y / z.Then one of −az − by, a z 2 − z b y 2 − y is nonzero, and one of σ 2 m , σ 2 2 m has e 1 ⊗ e 1 in its support.For more details, see 3 .