Tensor Products of the Gassner Representation of The Pure Braid Group
نویسنده
چکیده
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.
منابع مشابه
Irreducibility of the tensor product of Albeverio's representations of the Braid groups $B_3$ and $B_4$
We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$. We specialize the indeterminates used in defining these representations to non zero complex numbers. We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$. We then determine necessary and sufficient conditions that guarantee the irreducibility of th...
متن کاملOn the Kernel of the Gassner Representation
We study the Gassner representation of the pure braid group Pn by considering its restriction to a free subgroup F . The kernel of the restriction is shown to lie in the subgroup [ΓF,ΓF ], sharpening a result of Lipschutz.
متن کاملThe Gassner Representation for String Links
The Gassner representation of the pure braid group to GLn(Z[Z ]) can be extended to give a representation of the concordance group of n-strand string links to GLn(F ), where F is the field of quotients of Z[Z ], F = Q(t1, · · · , tn); this was first observed by Le Dimet. Here we give a cohomological interpretation of this extension. Our first application is to prove that the representation is h...
متن کاملOn Injective Homomorphisms for Pure Braid Groups, and Associated Lie Algebras
The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin’s pure braid group, a Lie algebra analyzed in work of Kohno [12, 13, 14], and Falk-Randell [9]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criteri...
متن کاملALEXANDER INVARIANTS OF ARRANGEMENTS 3 Our results
Let A be an arrangement of n complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism, : Fs ! Pn. Using the Gassner representation of the pure braid group, we nd an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a c...
متن کامل