The structure of twisted power partial isometries

Let $n>1$ and let $\{U_{ij}{\}}_{1\le i<j\le n}$ be $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ commuting unitaries on a Hilbert space $\mathcal{H}$. Suppose $U_{ji}:=U_{ij}^{\ast }$, $1\le i<j\le n$. An n-tuple of power partial isometries $(V_1,\mathrm{.}\mathrm{.}\mathrm{.},V_n)$ on Hilbert space $\mathcal{H}$ is called ${\mathcal{U}}_n$-twisted power partial isometry with respect to $\{U_{ij}{\}}_{i<j}$ (or simply ${\mathcal{U}}_n$-twisted power partial isometry if $\{U_{ij}{\}}_{i<j}$ is clear from the context) if $V_i^{\ast }{V}_j=U_{ij}{V}_j{V}_i^{\ast },V_i{V}_j=U_{ji}{V}_j{V}_i\text{ and }{V}_k{U}_{ij}=U_{ij}{V}_k(i,j,k=1,2,\mathrm{.}\mathrm{.}\mathrm{.},n,\text{ and }i\mathrm{\ne }j)\mathrm{.}$ We prove that each ${\mathcal{U}}_n$-twisted power partial isometry admits a Halmos and Wallen \cite{HW70} type orthogonal decomposition.