Higher Dimensional Irreducible Representations of the Pure Braid Group

Mohammad N. Abdulrahim, Hassan A Tarraf

Abstract


The reduced Gassner representation is a multi-parameter representation of $%
P_{n},$ the pure braid group on n strings. Specializing the parameters $%
t_{1},t_{2},...,t_{n}$ to nonzero complex numbers $x_{1},x_{2},...,x_{n}$
gives a representation $G_{n}(x_{1},\ldots ,x_{n}):P_{n}\rightarrow
GL(\mathbb{C}^{n-1})$ which is irreducible if and only if $x_{1}\ldots
x_{n}\neq 1$. In a previous work, we found a sufficient condition for the
irreducibility of the tensor product of two irreducible Gassner
representations. In our current work, we find a sufficient condition that guarantees the irreducibility of the tensor product of three Gassner representations. Next, a generalization of our result is given by considering the irreducibility of the tensor product of $k$ representations (\;$k \geq 3\;$).

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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