The Hilbert series of a linear symplectic circle quotient

TL;DR

This paper calculates the Hilbert series for symplectic quotients from circle actions, providing explicit formulas and analyzing the positivity and relations of Laurent coefficients, with implications for invariant theory.

Contribution

It introduces explicit formulas for Laurent coefficients of Hilbert series in symplectic quotients, including degenerate weights, and explores their positivity and relations.

Findings

Laurent coefficients are strictly positive.

Explicit formulas relate coefficients to weights.

Empirical evidence suggests general relations among coefficients.

Abstract

We compute the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation. Additionally, we elaborate explicit formulas for the lowest coefficients of the Laurent expansion of such a Hilbert series in terms of rational symmetric functions of the weights. Considerable efforts are devoted to including the cases where the weights are degenerate. We find that these Laurent expansions formally resemble Laurent expansions of Hilbert series of graded rings of real invariants of finite subgroups of $\U_n$. Moreover, we prove that certain Laurent coefficients are strictly positive. Experimental observations are presented concerning the behavior of these coefficients as well as relations among higher coefficients, providing empirical evidence that these relations hold in general.