Research Article | Open Access
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
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 .
Let be the braid group on strings. We consider the linear representation of called the Burau representation , 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 . 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 ).
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.
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 .
Lemma 3.1. (a) 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 3.3. 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 .
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 .
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 .
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 .
Claim 7. All tensors of the form and those obtained by permuting the indices are in .
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 .
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.
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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.