Minimal Nilpotent Orbits and Toric Varieties

Published 4 Nov 2025 in math.AG, math.AC, and math.RT | (2511.02179v1)

Abstract: Let $\overline{\mathcal{O}}\textrm{min} \cap (\mathfrak n⁺ \oplus \mathfrak n^-)$ be the collection of elements of $\mathfrak{sl}{n+1}(\mathbb C)$ with rank less than or equal to $1$ and with all diagonal entries equal to zero. We show that the coordinate ring $\mathbb C[\overline{\mathcal{O}}\textrm{min} \cap (\mathfrak n⁺ \oplus \mathfrak n^-)]$ of the scheme-theoretic intersection $\overline{\mathcal{O}}\textrm{min} \cap (\mathfrak n⁺ \oplus \mathfrak n^-)$ has a flat degeneration to the ring of $(\mathbb C^{{\times})^{n$-equivariant}} cohomology of the projective toric variety associated with the fan of compatible subsets of almost positive roots of type $C_n$. Then we compute the Hilbert series of $\mathbb C[\overline{\mathcal{O}}\textrm{min} \cap (\mathfrak n⁺ \oplus \mathfrak n^-)]$ and prove that $\overline{\mathcal{O}}\textrm{min} \cap (\mathfrak n⁺ \oplus \mathfrak n^-)$ is reduced and Gorenstein. Moreover, our proof method allows us to prove that the scheme-theoretic intersection $\overline{\mathcal{O}}_\textrm{min} \cap \mathfrak n^+$, of which the irreducible components are known as the ``orbital varieties'', is reduced and Cohen-Macaulay.