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

## TL;DR

This paper constructs $
$-supergrassmannians as homogeneous $
$-superspaces by gluing superdomains, describing the action of the super Lie group $GL( extbf{m})$, and proving transitivity and local chart gluing.

## Contribution

It provides an explicit construction of $
$-supergrassmannians as homogeneous superspaces with detailed group action descriptions.

## Key findings

- Constructed $
$-supergrassmannians via superdomain gluing.
- Described the $GL( extbf{m})$ action explicitly in functor of points.
- Proved the transitivity of the group action and local chart gluing.

## Abstract

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.