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General Linear and Symplectic Nilpotent Orbit Varieties

The condition of nilpotency is studied in the general linear Lie algebra $\mathfrak{gl}_{n}(\mathbb{K})$ and the symplectic Lie algebra $\mathfrak{sp}_{2m}(\mathbb{K})$ over an algebraically closed field of characteristic 0. In particular, the conjugacy class of nilpotent matrices is described through nilpotent orbit varieties $\mathcal{O}_λ$ and an algorithm is provided for computing the closure $\overline{\mathcal{O}_λ} \cong \text{Spec}\left(\mathbb{K}[X]\big/J_λ\right).$ We provide new generators for the ideal $J_λ$ defining the affine variety $\overline{\mathcal{O}_λ}$ which show that the generators provided in [J.Weyman - "The equations of conjugacy classes of nilpotent matrices", 1989] are not minimal. Furthermore, we conjecture the existence of local weak Néron models for nilpotent orbit varieties based on bounding $p$ in the polynomial ring with p-adic integer coefficients for which the equations defining $\mathcal{O}_λ$ can embed.