Smooth torus quotients of Richardson varieties in the Grassmannian

TL;DR

This paper investigates the geometric invariant theory (GIT) quotients of Richardson varieties in the Grassmannian under a torus action, providing combinatorial criteria for the smoothness of these quotients.

Contribution

It establishes necessary and sufficient combinatorial conditions for the smoothness of torus quotients of Richardson varieties in the Grassmannian.

Findings

Derived criteria for smoothness of quotients

Connected combinatorial conditions to geometric properties

Enhanced understanding of GIT quotients in algebraic geometry

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 $X^v_w$ in the Grassmannian $\mathrm{Gr}_{k,n}$ by $T$ with respect to a $T$-linearised line bundle $\cal{L}$ corresponding to the Pl\"{u}cker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety $T \backslash\mkern-6mu\backslash (X_w^v)^{ss}_T({\cal L})$ to be smooth.