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

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 $\mathrm{SL}_3(\mathbb{C})$.