$\mathrm{SO}(3)$-homogeneous decomposition of the flag scheme of $\mathrm{SL}_3$ over $\mathbb{Z}\left[1/2\right]$

TL;DR

This paper constructs and analyzes $ ext{SO}(3)$-homogeneous decompositions of the flag scheme of $ ext{SL}_3$ over $ ext{Spec}( ext{Z}[1/2])$, providing integral models of orbit structures.

Contribution

It introduces $ ext{Z}[1/2]$-forms of $ ext{SO}(3)$-orbits in the flag variety of $ ext{SL}_3$, establishing a $ ext{Z}[1/2]$-form of the orbit decomposition.

Findings

Established $ ext{Z}[1/2]$-forms of $ ext{SO}(3)$-orbits.

Proved these forms give a decomposition of the flag variety.

Extended orbit decomposition to integral models over $ ext{Z}[1/2]$.

Abstract

In this paper, we give $\mathbb{Z}\left[1/2\right]$-forms of $\mathrm{SO}(3,\mathbb{C})$-orbits in the flag variety of $\mathrm{SL}_3(\mathbb{C})$. We also prove that they give a $\mathbb{Z}\left[1/2\right]$-form of the $\mathrm{SO}(3,\mathbb{C})$-orbit decomposition of the flag variety of $\mathrm{SL}_3(\mathbb{C})$.