Krammer's representation of the pure braid group \(P_3\). (Q1958087)

Summary: We consider Krammer's representation of the pure braid group on three strings: \(P_3\to\text{GL}(3,Z[t^{\pm 1},q^{\pm 1}])\), where \(t\) and \(q\) are indeterminates. As it was done in the case of the braid group, \(B_3\), 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 Krammer's representation of the pure braid group, \(P_3\).