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

Abstract

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