Universal formula for Hilbert series of minimal nilpotent orbits

TL;DR

This paper derives a universal formula for the Hilbert series of minimal nilpotent orbit varieties across all simple Lie algebras, expressed via Vogel's parameters and hypergeometric functions.

Contribution

It introduces a universal formula for the Hilbert series of minimal nilpotent orbits applicable to all simple Lie algebras, unifying their descriptions.

Findings

Hilbert series expressed as a hypergeometric function ${}_{4}F_{3}$

Universal formula in terms of Vogel's parameters

Degree of the variety derived from the formula

Abstract

We show that the Hilbert series of the projective variety $X=P(O_{min}),$ corresponding to the minimal nilpotent orbit $\mathcal O_{min},$ is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel's parameters $\alpha,\beta,\gamma$ and represents a special case of the generalized hypergeometric function ${}_{4}F_{3}.$ A universal formula for the degree of $X$ is then deduced.