The Orbits of the Symplectic Group on the Flag Manifold

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 Gröbner 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.

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