Abstract
The reduced Burau representation is a one-parameter representation of , the braid group on strings. Specializing the parameter to nonzero complex number gives a representation : , which is either irreducible or has an irreducible composition factor : . In our paper, we let , and we determine a sufficient condition for the irreducibility of the tensor product of irreducible Burau representations. This is a generalization of our previous work concerning the cases and .
1. Introduction
Let be the braid group on strings. We consider the linear representation of called the Burau representation [1], which has a composition factor, the reduced Burau representation where is an indeterminate. Specializing , where , defines a representation which is either irreducible or has an irreducible subrepresentation of degree . 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 irreducible representations of the braid group, , irreducible? The question was answered in the cases and . We prove that the tensor product of an irreducible or with an irreducible or is irreducible if and only if . For more details, see [3]. We also consider the case and find a sufficient condition that guarantees the irreducibility of three irreducible representations of . In other words, we show that the tensor product of an irreducible or with an irreducible or with an irreducible or is irreducible if , , , , , , and . We fall short of finding a necessary and sufficient condition in the case (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 irreducible representations of the braid group, where . Therefore, our paper is concerned with the tensor product of the following irreducible representations:
Our main result is that, for , the above representations are irreducible if , for , where and are positive words of lengths and respectively, and do not have any in common.
2. Definitions
The braid group on strings, , is defined as an abstract group with generators ( and relations:(i) for ,(ii)if.
The generators are called the standard generators. Let be an indeterminate, and let be a Laurent polynomial ring over the complex numbers. All modules are vector spaces, so modules and modules will mean the same. We define the following representations of by matrices over .
Definition 1. The reduced Burau representation is given by
Definition 2. and .
Direct calculations show that
We identify with column vectors, we let denote the standard basis for , and we consider matrices to act by left multiplication on column vectors.
Definition 3. If , the support of , denoted by , is the set . If , the support of , also denoted by , is the set , and is called the coefficient of in .
Definition 4. For .
3. Preliminaries
Lemma 3.1. (a) for ,
for ,
for ,
for , , ,
(b) for .
Note that the above lemma remains true for any specialization , where .
Notation 1. Fix nonzero complex numbers . Let and , then we write .
Notation 2. Given , by a positive word of length , we mean a word that is written as a product of , where the number of s involved is , and their exponents are ones. Each in appears exactly once. We denote the word by . As an example, we write to stand for a word of length 3.
Lemma 3.2.
Lemma 3.3. Let be a specialization of the Burau representation making into a -module, where , then (a)let be the kernel of the homomorphism induced by (the augmentation ideal), then is equal to the vector space spanned by ,(b)if is a nonzero -submodule of , then . Hence, is the unique minimal nonzero -module of ,(c)if is not a root of , then , and is irreducible. If is a root of , then .
Hence, is its unique minimal nonzero -submodule. Of course, when is irreducible, but when is reducible, is the subrepresentation .
The main technical result is Proposition 4.1, which says that if for where and are positive words that do not have any in common, then is the unique minimal nonzero -module of . 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 Irreducible Representations
We obtain a result concerning the irreducibility of the tensor product of irreducible representations of the braid group, , where . We state our proposition and give an outline of a proof that goes along the same lines as in the cases and . However, a more general adequate proof is required here, which could be easily verified in the cases and 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.
Proposition 4.1. Let be a nonzero submodule of under the action of , where . Suppose that for one has then, contains all for , and contains , where the action of on the th factor is induced by .
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 .
Claim 1. There exists such that or .
Proof of Claim 1. When , we set
First, we show that . Second, we let , where . Using some recursive argument, we show that there is an element or of the form
where , and at least one of is nonzero. If , we are done. If , then
The determinant
is nonzero, since . Then one of is nonzero, and one of , has in its support. For more details, see [3].
As for the general case , we follow the same argument as above. The above is generalized to a ( matrix whose rows are given by for .
Here, for and s are all positive words of length . Using the hypothesis, the determinant of the matrix is nonzero, and this gives that one of the coefficients in has for .
Claim 2. Suppose that for some , then .
Proof of Claim 2. Here, represents all words of length . By our hypothesis, the proof is done.
Claim 3. There exists at least one such that , where and is neither nor .
Proof of Claim 3. Knowing that , then . It follows that
The proof is completed by applying repeatedly to the expression above.
Claim 4. All the tensors for all .
Proof of Claim 4. By applying to the tensors obtained from Claim 3 and then applying repeatedly, the proof is done.
Claim 5. For , all tensors of the form and and those obtained by permuting the indices are in .
Proof of Claim 5. Knowing that we apply induction on , and the proof is finished.
Claim 6. All tensors and those obtained by permuting the indices are in for .
Proof of Claim 6. This is done by induction on and using Claim 5.
Claim 7. All tensors of the form and those obtained by permuting the indices are in .
Proof of Claim 7. Applying induction on and using Claim 6, the proof is completed.
Claim 8. All tensors are in for all .
Proof of Claim 8. This is done by induction on and using the previous claims.
Claim 9. for all .
Proof of Claim 9. By Claim 8, we have that all the tensors are in , Consider then Knowing that , the proof is completed.
We now get our main theorem.
Theorem 4.2. Consider the tensor products of the irreducible representations where and denote a specialization of the reduced Burau representation and the irreducible subrepresentation of Lemma 3.3(b), respectively. If, for , one has that then the above representations are irreducible.
Proof. The proof is along the same lines as in the special cases and . All of the above representations are subrepresentations of
By Proposition 4.1, is the unique minimal nonzero module of . In particular, it is an irreducible -module. By Lemma 3.3, the first factor corresponds to one of the representations or , the second factor corresponds to one of the representations or , and so on according to whether or not , are roots of . Hence, all the above representations can be identified with the -module , so they are irreducible.