Tensor Products of the Gassner Representation of The Pure Braid Group†

Abstract

Abstract: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.

Burau Maps and Twisted Alexander Polynomials

“…Remark 3.14. Squier [30] observed (via an algebraic computation) that the reduced Burau representation is unitary with respect to a skew-Hermitian form (see also Abdulrahim [1] for a similar observation concerning the Gassner representation). Using the homological description outlined in Example 3.13, it was later understood ( [11,20,34]) that the above-mentioned skew-hermitian form arises from an intersection pairing on D ∞ n .…”

Coloured tangles and signatures

“…In the case of colored braids, this functor gives back the graph of the Gassner representation, the horizontal reflection of a braid is its inverse, and the Maslov index of the graphs of unitary automorphisms γ −1 1 , id and γ 2 is one possible definition of the Meyer cocycle of γ 1 and γ 2 . Therefore, in the case of µ-colored braids, our theorem is exactly the expected multivariable extension of (1).…”

“…Therefore, given two braids α, β ∈ B n and a root of unity ω, one can consider the Meyer cocycle of the two unitary matrices B ω (α) and B ω (β). The main theorem of [11] is the equality (1) sign ω ( αβ) − sign ω ( α) − sign ω ( β) = −Meyer (B ω (α), B ω (β)) for all α, β ∈ B n and ω ∈ S 1 of order coprime to n. (These authors actually work with the braid group on infinitely many strands B ∞ , and obtain an equality valid for any ω of finite order; however, their proof does yield the finer result stated above.) Let us mention that this equality not only relates two very much studied objects in knot theory, but also gives a very efficient algorithm for the computation of the signature, as the Meyer cocycle is easy to calculate (and the signature of unlinks vanishes).…”

“…, ω µ ) → sign ω (L) on the µ-dimensional torus T µ (see subsection 2.2 below). The Burau representation has a multivariable extension as well, called the Gassner representation [2], which is unitary [1], and it is natural to wonder if (1) holds in this multivariable setting.…”

On the Irreducibility of Fourth Dimensional Tuba's Representation of the Pure Braid Group on Three Strands

“…Researchers gave a great value for representations of the pure braid group P n . M.Abdulrahim gave a necessary and sufficient condition for the irreducibility of the complex specialization of the reduced Gassner representation of the pure braid group P n [1]. In our work, we mainly consider the irreducibility criteria of Albeverio-Rabanovich representation of the pure braid group P 3 with dimension three.…”