We give a stratification of the GIT quotient of the Grassmannian $G_{ 2,n}$ modulo the normaliser of a maximal torus of $SL_{ n}(k)$ with respect to the ample generator of the Picard group of $G_{ 2,n}$. We also prove that the flag variety $GL_{ n}(k)/B_{ n}$ can be obtained as a GIT quotient of $GL_{ n+1}(k)/B_{ n+1}$ modulo a maximal torus of $SL_{ n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{ n+1}(k)/B_{ n+1}$.
Categories: math.AG math.RT