GIT quotient of minimal dimensional Schubert variety modulo a subtorus

Abstract

Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $\omega_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(n\omega_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $n\omega_{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w_{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(n\omega_{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w_{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r^{th}$ stage of an iterated projective space bundle over $\mathbb{P}^{q-1}$.