Torus orbit closures in the flag variety

TL;DR

This paper reviews historical and recent research on the topology, geometry, and combinatorics of torus orbit closures in the flag variety, highlighting developments since the 1980s.

Contribution

It synthesizes past work and presents new insights into the structure of torus orbit closures in the flag variety.

Findings

Analysis of the topology of orbit closures

Geometric properties of orbit closures

Combinatorial descriptions of orbit structures

Abstract

The study of torus orbit closures in the (complete) flag variety was initiated by Klyachko and Gelfand--Serganova in the mid-1980s, but it seems that not much has been done since then. In this chapter, we present some of the work by Klyachko and Gelfand--Serganova and our recent work on the topology, geometry, and combinatorics of torus orbit closures in the flag variety.