Smooth torus quotients of Richardson varieties in the Grassmannian

Abstract

Let $k$ and $n$ be positive coprime integers with $k<n$. Let $T$ denote the subgroup of diagonal matrices in $SL(n,\mathbb{C})$. We study the GIT quotient of Richardson varieties $X_w^v$ in the Grassmannian ${\mathrm{Gr}}_{k,n}$ by $T$ with respect to a $T$-linearised line bundle $\mathcal{L}$ corresponding to the Plücker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety $T\backslash \backslash (X_w^v{)}_T^{ss}(\mathcal{L})$ to be smooth.