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

Abstract

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an 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 SL 2 -or SU 2 -module as well its first three Laurent coefficients. Contents 1. Introduction 1 Acknowledgements 4 2. Background 4 3. The Hilbert Series of the Module of Covariants 5 3.1. The Multivariate Hilbert Series 7 3.2. The Univariate Hilbert Series 9 3.3. The Laurent Coefficients of the Hilbert series 10 4. Hilbert Series of the Graded Algebra of Regular Functions on SU 2 -Symplectic Quotients 16 Appendix A. Hilbert Series in Low Dimensions 17 Appendix B. Visualization of γ on 0 (V ) 20 References 20