On the Construction of $\mathbb{Z}_2^n-$Supergrassmannians as Homogeneous $\mathbb{Z}_2^n-$Superspaces

In this paper, we construct the $\mathbb Z_{2}^{n}-$supergrassmannians by gluing of the $\mathbb Z_{2}^{n}-$superdomains and give an explicit description of the action of the $\mathbb Z_{2}^{n}-$super Lie group $GL(\overrightarrow{\textbf{m}})$ on the $\mathbb Z_{2}^{n}-$supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the supergrassmannian.