Mohammad N. Abdulrahim

A faithfulness criterion for the Gassner representation of the pure braid group

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, Pn(n > 3). Long and Paton proved that if a Burau matrix M has ones on the diagonal and zeros below the diagonal then M is the identity matrix. In this paper, a generalization of Long and Paton’s result will be proved. Our main theorem is that if the trace of the image of an element of Pn under the reduced Gassner representation is n− 1, then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.

Complex specializations of the reduced Gassner representation of the pure braid group

We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation Gn(z) : Pn → GLn−1(C) to be irreducible. It will be shown that for z = (z1, . . . , zn) ∈ (C∗)n, Gn(z) is irreducible if and only if z1 . . . zn 6= 1.

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.

ON THE COMPOSITION OF THE BURAU REPRESENTATION AND THE NATURAL MAP Bn → Bnk

A lot of linear representations of the braid group, Bn, arise as a result of treating braids as automorphisms of a free group. In this paper, we consider the composition of F. R. Cohens map Bn → Bnk and the embedding Bnk → Aut (Fnk). This gives us a linear representation of Bn whose composition factors are one copy of the Burau representation and k - 1 copies of the standard representation, a representation investigated by I. Sysoeva.

Tensor Products of the Gassner Representation of The Pure Braid Group

The reduced Gassner representation is a multi-parameter representation of Pn, the pure braid group on n strings. Specializing the parameters t1, t2,...,tn to nonzero complex numbers x1,x2,...,xn gives a representation Gn(x1,...,xn): Pn GL( n 1 ) which is irreducible if and only if x1...xn 1.We find a sufficient condition that guarantees that the tensor product of an irreducible Gn(x1,...,xn)with an irreducible Gn(y1, ..., yn) is irreducible. We fall short of finding a necessary and sufficient condition for irreducibility of the tensor product. Our work is a continuation of a previous one regarding the tensor product of complex specializations of the Burau representation of the braid group.

WADA'S REPRESENTATION IS OF BURAU TYPE

We consider the Magnus representation of the image of the braid group under a representation discovered by Wada. We first investigate the question about the reducibility of this representation and show that it is of Burau type. Next, we show that the images of the generators of the braid group under that representation are unitary relative to a hermitian matrix. This is similar to the well known result by Squier that asserts that the Burau representation of the braid group is unitary.

A New Six Dimensional Representation of the Braid Group on Three Strands and its Irreducibility and Unitarizability

We consider the braid group on three strands, B3 and construct a complex valued representation of it with degree 6, namely, ρ : B3 → GL6(C). First, we show that this representation is irreducible and not equivalent to either Burau or Krammer’s representations. Second, we prove that the representation is unitary relative to an invertible hermitian matrix.

Krammer's Representation of the Pure Braid Group, 3

We consider Krammers representation of the pure braid group on three strings: 𝑃3→𝐺𝐿(3,𝑍[𝑡±1,𝑞±1]), where 𝑡 and 𝑞 are indeterminates. As it was done in the case of the braid group, 𝐵3, we specialize the indeterminates 𝑡 and 𝑞 to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammers representation of the pure braid group, 𝑃3.

Tensoring three irreducible linear representations of the braid group

The reduced Burau representation is a one-parameter representation of B n , the braid group on n strings. Specializing the parameter to a non-zero complex number x gives a representation β n (x) : B n → GL(ℂ n−1) which is either irreducible or has an irreducible composition factor . It was proved in a previous work that the tensor product of an irreducible β n (x) or with an irreducible β n (y) or is irreducible unless x = y ±1. In our work, we will consider three (irreducible) specializations of the Burau representations and we will find conditions on the non-zero complex numbers x, y and z that guarantee the irreducibility of the tensor product of the three linear representations β n (x) or , β n (y) or and β n (z) or .

Tuba’s Representation of the Pure Braid Group on Three Strands

We consider Tuba’s representation of the pure braid group, P3, defined by the map ψ : P3 −→ GL (V ), where V is an algebraically closed field. We then specialize the indeterminates used in defining the representation to nonzero complex numbers. Our objective is to find necessary and sufficient conditions that guarantee the irreducibility of Tuba’s representations of the pure braid group P3 with dimensions d = 2 and d = 3.