  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.
