GENERALIZATIONS OF THE STANDARD ARTIN REPRESENTATION ARE UNITARY

We consider the Magnus representation of the image of the braid group under the generalizations of the standard Artin representation discovered by M. Wada. We show that the images of the generators of the braid group under the Magnus representation are unitary relative to a Hermitian matrix. As a special case, we get that the Burau representation is unitary, which was known and proved by C. C. Squier.


Introduction
The braid group B n has a well-known representation due to Artin in the group Aut(F n ) of automorphisms of the free group F n generated by x 1 ,...,x n .The automorphism corresponding to the braid generator σ i takes x i to x i x i+1 x i −1 ; x i+1 to x i , and fixes all other free generators.Such a representation of the braid group by automorphisms of a free group was proved to be faithful [3, page 25].
In Section 2, we present an infinite series of representations generalizing the standard Artin representation, which were discovered by Wada [8].More precisely, for an arbitrary nonzero integer k, the automorphism corresponding to the braid generator σ i takes x i to x i k x i+1 x i −k ; x i+1 to x i , and fixes all other free generators.Shpilrain has shown that these representations are indeed faithful [6, page 773].
In Section 3, after having defined the automorphism corresponding to the braid generator, suggested by Wada, we apply the Magnus representation to these subgroups of Aut(F n ) to get linear irreducible representations B n → GL n−1 (C[t ±1 ]).We show that for any nonzero integer k, the linear representations obtained are unitary relative to a Hermitian matrix.In particular, this shows that the Burau representation, namely when k = 1, is conjugate to an ordinary unitary representation; which was proved by Squier [7].
Showing that Wada's representations are unitary might possibly help us to determine whether or not such matrix representations of the braid group are faithful.A similar argument was done in the case of the standard Artin representation (see [1, page 1257]).It was known that for k = 1, the Burau representation is not faithful for n ≥ 6 [5].It is now known that the Burau representation for n = 5 is not faithful [2].

Definitions
The braid group on n strings, B n , is the abstract group with generators σ 1 ,...,σ n−1 and a presentation as follows: (2.1) According to the standard Artin representation, the automorphism corresponding to σ i sends x i to x i x i+1 x i −1 ; x i+1 to x i , and fixes all other free generators.
Definition 2.1.The generalizations of the standard Artin representation, discovered by Wada, assert that the automorphism corresponding to σ i takes (2.2) By applying the Magnus representation to the image of the braid group under the generalization of the standard Artin representation, we determine the linear representations [3].The automorphism σ i is mapped onto the n × n matrix which differs from the identity only by a 2 × 2 block with the top-left corner in the (i,i)th place.More precisely, It is clear that the subspace generated by the column vector (1,1,...,1) T is invariant under this representation, where T is the transpose.Therefore, these representations, for different values of k, are reducible. ) These representations are irreducible by [4,Theorem 5].Notice that the representation φ 1 is (conjugate to) the reduced Burau representation of the braid group as presented in [4].

Wada's representations are unitary
Notation 3.1.Let ( * ) : M m (C[t ±1 ]) be an involution defined as follows: Now define the following (n − 1) × (n − 1) matrix, M, in a way that each column looks like (0,...,0,−t k ,t k + 1,−1,0,...,0) T , where t k + 1 is a diagonal entry and T is the transpose.More precisely, we have For simplicity, we denote the matrix φ k (σ i ) corresponding to the braid generator, σ i , under Wada's representations, by X k,i , where X k,i = I n−1 − A i B i , where A i , B i are given by Definition 2.2.
We now prove our main theorem.
Theorem 3.3.The images of the generators of B n under Wada's representations, φ k , are unitary relative to M, that is, for , where u 2 = t.Let N = u −k M, then by direct substitution, we get It is clear that N is Hermitian (N * = N) and X k,i N(X k,i ) * = N. Next, our objective is to show that a certain specialization N of N is equivalent to the identity matrix in some extension field, that is, for some matrix U, we have that From linear algebra, it is well known that a Hermitian matrix is positive definite if and only if each of the principal minors is positive.In that case, the matrix will be equivalent to the identity matrix.
The principal minors of N are of the form det(D m ), where 1 ≤ m ≤ n − 1 and D m is an m × m matrix (upper-left corners of N).It is then easy to see the following lemma.(3.9) Proof.By induction on m, we get Let u = a, where a is a complex number lying in an open arc around 1 on the unit circle.By having an explicit formula for the principal minors of N as in Lemma 3.4, it is then possible to completely determine the arc around 1 where a belongs to.The choice of this arc depends on the values of k and n.Along the same lines as in [1, pages 1254-1255], we can easily get the following lemma.Hence, the matrix N is a positive definite Hermitian matrix under the complex specialization u = a belonging to the open arc bounded by e −πi/kn and e πi/kn .We denote this matrix by N. By a theorem in linear algebra, there exists a matrix U such that As in [1, page 1255], the next theorem shows that a conjugate of Wada's representation is unitary.Here, a matrix X is unitary if XX * = X * X = I.
Theorem 3.6.The complex specialization of Wada's representation of B n (having t = u 2 = a 2 and a is around 1) is conjugate to an ordinary unitary representation.
Proof.Consider the composition map Let f (X k,i ) be the image of X k,i under the complex specialization u = a, where a lies in an arc around 1 bounded by e −πi/kn and e πi/kn .
Having that N = UU * , we let Notice that, under the case k = 1, Theorem 3.6 implies that the specialization of the Burau representation is conjugate to an ordinary unitary representation; which was proved by Squier [7].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Lemma 3 . 5 .
Let a be a complex number on the unit circle.Then det(D m ) is positive for all m = 1,2,...,n − 1 if and only if a lies in an open arc around 1 bounded by e −πi/kn and e πi/kn .