Exponential Hilbert series of equivariant embeddings

TL;DR

This paper investigates the exponential Hilbert series of equivariant projective varieties under semisimple algebraic groups, establishing relationships with geometric properties and deriving combinatorial identities involving Stirling numbers.

Contribution

It introduces a new relationship between exponential Hilbert series and geometric invariants, and derives novel combinatorial identities for their coefficients.

Findings

Established a link between exponential Hilbert series, degree, and dimension of varieties.

Derived a combinatorial identity for polynomial coefficients of the series.

Applied identities to connect with Stirling numbers of the first and second kinds.

Abstract

In this article, we study properties of the exponential Hilbert series of a $G$-equivariant projective variety, where $G$ is a semisimple, simply-connected complex linear algebraic group. We prove a relationship between the exponential Hilbert series and the degree and dimension of the variety. We then prove a combinatorial identity for the coefficients of the polynomial representing the exponential Hilbert series. This formula is used in examples to prove further combinatorial identities involving Stirling numbers of the first and second kinds.