# Torus quotients of Richardson varieties in the Grassmannian

## TL;DR

This paper investigates the geometric invariant theory (GIT) quotients of certain Schubert and Richardson varieties in the Grassmannian, establishing smoothness conditions and providing new combinatorial proofs for projective normality.

## Contribution

It demonstrates smoothness of GIT quotients when gcd(r,n)=1 and offers a novel combinatorial proof of projective normality for GIT quotients of Grassmannian varieties.

## Key findings

- GIT quotient is smooth when gcd(r,n)=1.
- Provides a new combinatorial proof of projective normality.
- Studies GIT quotients of Richardson varieties in specific cases.

## Abstract

We study the GIT quotient of the minimal Schubert variety in the Grassmannian admitting semistable points for the action of maximal torus $T$, with respect to the $T$-linearized line bundle ${\cal L}(n \omega_r)$ and show that this is smooth when $gcd(r,n)=1$. When $n=7$ and $r=3$ we study the GIT quotients of all Richardson varieties in the minimal Schubert variety. This builds on previous work by Kumar \cite{kumar2008descent}, Kannan and Sardar \cite{kannan2009torusA}, Kannan and Pattanayak \cite{kannan2009torusB}, and recent work of Kannan et al \cite{kannan2018torus}. It is known that the GIT quotient of $G_{2,n}$ is projectively normal. We give a different combinatorial proof.