General Linear and Symplectic Nilpotent Orbit Varieties

Abstract

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 ${\mathcal{O}}_{\lambda }$ and an algorithm is provided for computing the closure $\overline{{\mathcal{O}}_{\lambda }}\cong \text{Spec}\left(\mathbb{K}[X]/J_{\lambda }\right).$ We provide new generators for the ideal $J_{\lambda }$ defining the affine variety $\overline{{\mathcal{O}}_{\lambda }}$ 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 ${\mathcal{O}}_{\lambda }$ can embed.

9 pages, 1 figure