\({{\,\textrm{SO}\,}}(3)\)-Homogeneous Decomposition of the Flag Scheme of \({{\,\textrm{SL}\,}}_3\) over \(\mathbb {Z}\left[ 1/2\right] \)

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