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

Abstract

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