The Hilbert series and $a$-invariant of circle invariants

TL;DR

This paper derives explicit formulas for the Hilbert series and $a$-invariant of circle invariants in polynomial algebras, revealing their structure and conditions for Gorenstein properties based on weight vectors.

Contribution

It provides new explicit formulas for the Hilbert series and $a$-invariant of circle invariants, including handling singularities and symmetry properties, advancing understanding of their algebraic structure.

Findings

Explicit formulas for Hilbert series coefficients

Identification of Gorenstein and non-Gorenstein cases

Use of partial Schur polynomials in invariant theory

Abstract

Let $V$ be a finite-dimensional representation of the complex circle $\mathbb{C}^\times$ determined by a weight vector $\mathbf{a}\in\mathbb{Z}^n$. We study the Hilbert series $\operatorname{Hilb}_{\mathbf{a}}(t)$ of the graded algebra $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ of polynomial $\mathbb{C}^\times$-invariants in terms of the weight vector $\mathbf{a}$ of the $\mathbb{C}^\times$-action. In particular, we give explicit formulas for $\operatorname{Hilb}_{\mathbf{a}}(t)$ as well as the first four coefficients of the Laurent expansion of $\operatorname{Hilb}_{\mathbf{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomial that are independently symmetric in two sets of variables. We similarly give an explicit formula for the $a$-invariant of $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.