The Hilbert Series of a Linear Symplectic Circle Quotient

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 effort is devoted to including the cases in which 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 . 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.

2000 AMS Subject Classification:

• Primary 53D20
• 13A50
• Secondary 11F2

Keywords:

• singular symplectic reduction
• invariant theory
• circle actions

Acknowledgments

We express our gratitude to Leonid Bedratyuk for assistance in computing invariants of . The second author would like to thank the Centre for the Quantum Geometry of Moduli Spaces for hospitality during the completion of this manuscript.

FUNDING

The research of the first author was supported by the Centre for the Quantum Geometry of Moduli Spaces, which is funded by the Danish National Research Foundation, and by the Austrian Ministry of Science and Research BMWF, Start-Prize Y377.

Notes

¹Available at faculty.rhodes.edu/seaton/symp_red/ (use the link to the file HilbertSeriesS1.nb).

²Available respectively at wolfram.com/mathematica/ , singular.uni-kl.de , mathematik.uni-osnabrueck.de/normaliz/ , and math.uiuc.edu/Macaulay2/.