The Hilbert Series of a Linear Symplectic Circle Quotient

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...

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Bibliographic Details
Published in:Experimental mathematics 2014-01, Vol.23 (1), p.46-65
Main Authors: Herbig, Hans-Christian, Seaton, Christopher
Format: Article
Language:English
Subjects:
circle actions
invariant theory
Primary 53D20
Secondary 11F2
singular symplectic reduction
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Summary: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.
ISSN:1058-6458
1944-950X
DOI:10.1080/10586458.2013.863745