Orbits structures and complexity in Schubert and Richardson varieties
ABSTRACT
The study of orbits and orbit closures in the flag variety has a long and storied history with deep connections to algebraic combinatorics, Lie theory, and representation theory. The orbits of Borel subgroups and their Zariski closures, the Schubert varieties, have been of particular import. A central notion in this area is the complexity of a reductive group action on a variety, which equals the minimum codimension of a Borel subgroup orbit. In this talk 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. Then, when a Levi subgroup acts on a Schubert variety, we exhibit a codimension preserving bijection between the Levi-Borel subgroup orbits in the big open cell of that Schubert variety and the torus orbits in the big open cell of a distinguished Schubert subvariety. This allows us to give a type-uniform formula for the Levi-Borel complexity of that Schubert variety.
