On the composition of the Cohen map and the Tong–Yang–Ma representation

We study the composition of F. R. Cohen map \(B_n\rightarrow B_{n k}\) $B_n\to B_{\mathit{nk}}$ with the Tong-Yang-Ma representation \(B_{nk}\rightarrow GL_{nk}(\mathbb {Z}[t^{\pm 1}])\) $B_{\mathit{nk}}\to G{L}_{\mathit{nk}}(\mathbb{Z}[t^{\pm 1}])$ in the case \(k=2\) $k=2$ , where \(B_n\) $B_n$ is the braid group on n strings. We prove that the transpose of the representation thus obtained of degree 2n is the direct sum of two copies of Tong-Yang-Ma representations, each of degree n, with t replaced by \(t^2\) $t^2$ .