On the irreducibility of the extensions of Burau and Gassner representations

Let \(Cb_n\) be the group of basis conjugating automorphisms a free \(\mathbb {F}_n\), and \(C_n\) {F}_n\). Valerij G. Bardakov has constructed representations \(Cb_n\), in groups \(GL_n(\mathbb {Z}[{t_1}^{\pm 1}, \ldots ,{t_n}^{\pm 1}])\) {Z}[{t}^{\pm respectively, where \(t_1, , t_n, t\) are indeterminate variables. We show that these reducible we determine irreducible components {C})\), which obtained by giving values to variables above. Next, consider tensor product study their irreduciblity case \(n=3\).