Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $S{L}_n(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $G{L}_n(k)\mathrm{/}{B}_n$ can be obtained as a GIT quotient of $G{L}_{n+1}(k)\mathrm{/}{B}_{n+1}$ modulo a maximal torus of $S{L}_{n+1}(k)$ for a suitable choice of an ample line bundle on $G{L}_{n+1}(k)\mathrm{/}{B}_{n+1}$.