Torus quotients of Richardson varieties in $G_{r,qr+1}$ (2310.11091v1)

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 $w_{r,n}(n\omega_r) \leq 0$ (respectively, $v_{r,n}(n\omega_r) \geq 0$). Let $v \leq v_{r,n}$ and $X^v_{w_{r,n}}$ 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 $X^{v}{w{r,n}}$ for the action of $T$ is the product of projective spaces with respect to the descent of the line bundle $\mathcal{L}(n\omega_r).$