B_{n-1}-bundles on the flag variety, I
Mark Colarusso, Sam Evens

TL;DR
This paper establishes a recursive method to classify B_{n-1}-orbits on flag varieties for GL(n) and SO(n), revealing their bundle structure and paving the way for explicit combinatorial classifications.
Contribution
It introduces a uniform inductive approach to classify B_{n-1}-orbits on flag varieties for GL(n) and SO(n), extending previous work and enabling explicit orbit descriptions.
Findings
Orbits form bundles over smaller flag varieties.
Develops an inductive classification procedure.
Lays groundwork for explicit combinatorial orbit classification.
Abstract
We show that each orbit of a Borel subgroup of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a -orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify -orbits on the flag variety. Our method is essentially uniform in the two cases. As further consequences, in the sequel to this paper we give an explicit combinatorial classification of orbits and determine completely the closure relation between orbit closures. This further develops work of Hashimoto in the general linear group case.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Geometric and Algebraic Topology
