On Orbifold Criteria for Symplectic Toric Quotients

Farsi, Carla

doi:10.3842/sigma.2013.032

Cited by 14 publications

(44 citation statements)

References 27 publications

(42 reference statements)

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“…In Section 3 we use Proposition 1.4 to show Theorem 1.6. The latter is used in Section 4 to give a proof of our main result, Theorem 1.3, that is based on Molien's formula and the fact that a moment map of a faithful torus representation forms a regular sequence in the ring of invariants [7,5]. In Section 5 we deduce from Corollary 1.5 an identity for the coefficients of the even Euler polynomials.…”

Section: Theorem 13 Conjecture 12 Holds If G Is a Torusmentioning

confidence: 95%

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: Theorem 13 Conjecture 12 Holds If G Is a Torusmentioning

confidence: 95%

On compositions with 𝑥²/(1-𝑥)

Herbig¹,

Herden²,

Seaton³

2015

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…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%

“…It is moreover pointed out in [2] that C ∞ (M 0 ) inherits a Poisson bracket {, } from C ∞ (V ). Hence the triple (M 0 , C ∞ (M 0 ), {, }) is a Poisson differential space; see [10,Definition 5]. Using flows of invariant functions, it is shown in [3,29] that the symplectic stratification can be reconstructed from the Poisson algebra (C ∞ (M 0 ), { , }).…”

Section: Introductionmentioning

confidence: 99%

“…A language to axiomatizes this idea has been suggested in [10]; the essentials we recall next. The algebra of real-valued regular functions R[V ] on V forms a Poisson subalgebra of C ∞ (V ).…”

Section: Introductionmentioning

confidence: 99%

“…The appropriate tool is provided by [10,Theorem 6], the so-called Lifting Theorem. It allows to lift certain (Z-graded) Poisson homomorphisms between Poisson algebras of regular functions to the Poisson algebras of smooth function, namely those that are compatible with the embeddings into eucildean space.…”

Section: Introductionmentioning

confidence: 99%

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