Torus Quotients of Richardson Varieties

TL;DR

This paper demonstrates that GIT quotients of specific Richardson varieties in Grassmannians, under a maximal torus action, are isomorphic to products of projective spaces, revealing a geometric structure of these quotients.

Contribution

It establishes that certain Richardson varieties' GIT quotients are explicitly describable as products of projective spaces, providing new geometric insights.

Findings

GIT quotients of Richardson varieties are products of projective spaces

The result applies to cases where (r,n)=1

Provides a geometric description of quotients in Grassmannians

Abstract

For $1\le r\le n-1,$ let $G_{r,n}$ denote the Grassmannian parametrizing $r$-dimensional subspaces of $\mathbb{C}^{n}.$ Let $(r,n)=1.$ In this article we show that the GIT quotients of certain Richardson varieties in $G_{r,n}$ for the action of a maximal torus in $SL(n,\mathbb{C})$ are the product of projective spaces with respect to the descent of a suitable line bundle.