Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$

TL;DR

This paper calculates the Hilbert series for symplectic quotients related to SU(2)-modules, providing explicit formulas, an algorithm for computation, and results for modules up to dimension 10.

Contribution

It introduces a method to compute the Hilbert series of symplectic quotients for SU(2)-modules and provides explicit formulas and coefficients for these series.

Findings

Derived explicit Hilbert series formulas for SU(2)-modules.

Developed an algorithm to compute Hilbert series for modules up to dimension 10.

Computed Laurent coefficients of the Hilbert series at t=1.

Abstract

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an $\operatorname{SU}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t = 1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary $\operatorname{Sl}_2$- or $\operatorname{SU}_2$-module as well its first three Laurent coefficients.