# Description of infinite orbits on multiple projective spaces

## TL;DR

This paper describes the orbit decomposition of multiple projective spaces under the diagonal action of the general linear group, revealing infinitely many orbits for four or more factors and providing representatives.

## Contribution

It provides a detailed description of orbit structures on multiple projective spaces under group actions, including the classification of infinitely many orbits for higher dimensions.

## Key findings

- Number of orbits is infinite for m ≥ 4.
- Explicit representatives of orbits are constructed.
- Orbit decomposition is fully characterized for the given group action.

## Abstract

Let $G$ be the general linear group of the degree $n\geq 2$ over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. In this article, we give a description of orbit decomposition of the multiple projective space $G^m/P^m$ under the diagonal action of $G$ where $P$ is the maximal parabolic subgroup of $G$ such that $G/P\cong\mathbb{P}^{n-1}\mathbb{K}$. We also construct representatives of orbits. If $m\geq 4$, the number of orbits is infinite, and we give a description of those uncountably many orbits.