Universal formula for the Hilbert series of minimal nilpotent orbits

Abstract

Abstract. We show that the Hilbert series of the projective variety X = P (Omin), corresponding to the minimal nilpotent orbit Omin, is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel's parameters α, β, γ and represents a special case of the generalized hypergeometric function 4F3. A universal formula for the degree of X is then deduced.

Exponential Hilbert series of equivariant embeddings

“…where a is the degree of the embedding of the projective variety and d is again the dimension of the variety (see [5], [8]). Computation of the Hilbert polynomial and/or the Hilbert series of A then yields geometric information about the variety, which can be computed in a very simple way.…”