Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

TL;DR

This paper studies geometric invariant theory quotients of Grassmannians and flag varieties related to the general linear group, providing stratifications and new quotient constructions with implications for symmetric group representations.

Contribution

It introduces a stratification of the GIT quotient of Grassmannians and constructs flag varieties as GIT quotients, revealing new geometric relationships.

Findings

Stratification of GIT quotient of G_{2,n} by the normaliser of a maximal torus.

Flag variety as a GIT quotient of a higher-dimensional flag variety.

Insights into the structure of symmetric group representations via geometric methods.

Abstract

We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.