Orbit structures and complexity in Schubert and Richardson Varieties

TL;DR

This paper provides a uniform formula for torus complexity in Richardson varieties, describes the torus action on Deodhar components, and establishes a bijection relating Levi subgroup orbits to torus orbits in Schubert varieties, extending complexity results.

Contribution

It introduces a type-uniform formula for torus complexity in Richardson varieties and describes the torus action on Deodhar components, extending Levi-Borel complexity results.

Findings

Explicit description of torus action on Deodhar components

Type-uniform formula for torus complexity in Richardson varieties

Bijection between Levi-Borel orbits and torus orbits in Schubert varieties

Abstract

The goal of this paper is twofold. Firstly, we provide a type-uniform formula for the torus complexity of the usual torus action on a Richardson variety, by developing the notion of algebraic dimensions of Bruhat intervals, strengthening a type $A$ result by Donten-Bury, Escobar and Portakal. In the process, we give an explicit description of the torus action on any Deodhar component as well as describe the root subgroups that comprise the component. Secondly, when a Levi subgroup in a reductive algebraic group acts on a Schubert variety, we exhibit a codimension preserving bijection between the Levi-Borel subgroup (a Borel subgroup in the Levi subgroup) orbits in the big open cell of that Schubert variety and torus orbits in the big open cell of a distinguished Schubert subvariety. This bijection has many applications including a type-uniform formula for the Levi-Borel complexity of the usual Levi-Borel subgroup action on a Schubert variety. We conclude by extending the Levi-Borel complexity results to a large class of Schubert varieties in the partial flag variety.