The Orbits of the Symplectic Group on the Flag Manifold

TL;DR

This paper studies the action of the complex symplectic group on the flag manifold, using algebraic and combinatorial methods to understand orbit structures and their closures.

Contribution

It introduces a concrete approach to analyze symplectic group orbits on flag manifolds through Gr"obner degenerations and identifies key orbit closure elements.

Findings

Orbit closures degenerate to unions of Schubert varieties

Identification of fundamental orbit closure elements

Provides a combinatorial framework for orbit analysis

Abstract

We examine the orbits of the (complex) symplectic group, $Sp_n$, on the flag manifold, $\mathscr{F}\ell(\mathbb{C}^{2n})$, in a very concrete way. We use two approaches: we Gr\"obner degenerate the orbits to unions of Schubert varieties (for a equations of a particular union of Schubert varieties see \cite{NWunions}) and we find a subset $\mathcal{A}$ of the orbit closures containing the basic elements of the poset of orbit closures under containment, which represent the geometric and combinatorial building blocks for the orbit closures.