Hilbert series of symplectic quotients by the 2-torus

TL;DR

This paper calculates the Hilbert series of real regular functions on symplectic quotients by the 2-torus, providing explicit formulas and an algorithm for computation.

Contribution

It introduces a method to compute the Hilbert series and Laurent coefficients for symplectic quotients by the 2-torus, including explicit examples.

Findings

Derived the Hilbert series for symplectic quotients by the 2-torus

Computed the first four Laurent coefficients at t=1

Provided an algorithm for explicit calculations

Abstract

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the $2$-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at $t = 1$. We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples.