Torus quotients of Richardson varieties in $G_{r,qr+1}$

Abstract

Let $r$ and $q$ be positive integers and $n=qr+1.$ Let $G=SL(n,\mathbb{C})$ and $T$ be a maximal torus of $G.$ Let $P^{{\alpha }_r}$ be the maximal parabolic subgroup of $G$ corresponding to the simple root ${\alpha }_r.$ Let ${\omega }_r$ be the fundamental weight corresponding to ${\alpha }_r.$ Let $W$ be the Weyl group of $G$ and $W_{P^{{\alpha }_r}}$ be the Weyl group of $P^{{\alpha }_r}.$ Let $W^{P^{{\alpha }_r}}$ be the set of all minimal coset representatives of $W/W_{P^{{\alpha }_r}}$ in $W.$ Let $w_{r,n}$ (respectively, $v_{r,n}$) be the minimal (respectively, maximal) element in $W^{P^{{\alpha }_r}}$ such that $w_{r,n}(n{\omega }_r)\le 0$ (respectively, $v_{r,n}(n{\omega }_r)\ge 0$). Let $v\le v_{r,n}$ and $X_{w_{r,n}}^v$ be the Richardson variety in $G_{r,n}$ corresponding to $v$ and $w_{r,n}.$ In this article, we give a sufficient condition on $v$ such that the GIT quotient of $X_{w_{r,n}}^v$ for the action of $T$ is the product of projective spaces with respect to the descent of the line bundle $\mathcal{L}(n{\omega }_r).$

23 pages, to appear in Proceedings-Mathematical Sciences