The Hilbert Series of a Linear Symplectic Circle Quotient

Herbig, Hans-Christian; Seaton, Christopher

doi:10.1080/10586458.2013.863745

Cited by 14 publications

(19 citation statements)

References 10 publications

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“…When the action is generic, i.e. no two weights have the same absolute value, an algorithm for computing the Hilbert series is described in [8,Section 4] and has been implemented on Mathematica [18]. To check that a concrete Hilbert series is symplectic of order d = dim(M 0 ), we use the substitution t → 1 2 y + 2 ± y(y + 4)…”

Section: Symplectic Quotients By Smentioning

confidence: 99%

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On compositions with 𝑥²/(1-𝑥)

Herbig¹,

Herden²,

Seaton³

2015

Proc. Amer. Math. Soc.

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Abstract. In the past, empirical evidence has been presented that Hilbert series of symplectic quotients of unitary representations obey a certain universal system of infinitely many constraints. Formal series with this property have been called symplectic. Here we show that a formal power series is symplectic if and only if it is a formal composite with the formal power series x 2 /(1 − x). Hence the set of symplectic power series forms a subalgebra of the algebra of formal power series. The subalgebra property is translated into an identity for the coefficients of the even Euler polynomials, which can be interpreted as a cubic identity for the Bernoulli numbers. Furthermore we show that a rational power series is symplectic if and only if it is invariant under the idempotent Möbius transformation x → x/(x − 1). It follows that the Hilbert series of a graded Cohen-Macaulay algebra A is symplectic if and only if A is Gorenstein with its a-invariant and its Krull dimension adding up to zero. It is shown that this is the case for algebras of regular functions on symplectic quotients of unitary representations of tori.

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Section: Symplectic Quotients By Smentioning

confidence: 99%

“…The main motivation for our investigation is Conjecture 1.2 below, that has been formulated in [8]. We recall the following definition from [8].…”

Section: Introductionmentioning

confidence: 99%

On compositions with 𝑥²/(1-𝑥)

Herbig¹,

Herden²,

Seaton³

2015

Proc. Amer. Math. Soc.

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“…Here, we specialize the discussion to unitary representations V of the circle G = S 1 and recall results from [16] regarding the Hilbert series of the N-graded algebra R[M 0 ] of regular functions on the symplectic quotient M 0 = Z/G. It will be convenient to identify V with C n by choosing coordinates z 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [10] for more details. In addition, we summarize the formulas from [16] for the first four coefficients of the Laurent series of a S 1 -reduced space and a linear symplectic orbifold for the cases we will need.…”

Section: Introductionmentioning

confidence: 99%

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An Impossibility Theorem for Linear Symplectic Circle Quotients

Herbig

Seaton

2015

Reports on Mathematical Physics

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7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

Iakovidis

Qiu

Rocén

et al. 2020

J. High Energ. Phys.

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We study 7D maximally supersymmetric Yang-Mills theory on 3-Sasakian manifolds. For manifolds whose hyper-Kähler cones are hypertoric we derive the perturbative part of the partition function. The answer involves a special function that counts integer lattice points in a rational convex polyhedral cone determined by hypertoric data. This also gives a more geometric structure to previous enumeration results of holomorphic functions in the literature. Based on physics intuition, we provide a factorisation result for such functions. The full proof of this factorisation using index calculations will be detailed in a forthcoming paper.

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The Hilbert Series of a Linear Symplectic Circle Quotient

Cited by 14 publications

References 10 publications

On compositions with 𝑥²/(1-𝑥)

On compositions with 𝑥²/(1-𝑥)

An Impossibility Theorem for Linear Symplectic Circle Quotients

7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

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