Residue formulae for vector partitions and Euler–MacLaurin sums

Szenes, András; Vergne, Michèle

doi:10.1016/s0196-8858(02)00538-9

Cited by 74 publications

(94 citation statements)

References 13 publications

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“…General partial fractions. The results we present from now on are just simple reformulations of the results by Szenes-Vergne [26]. We only stress more the role of no-broken circuits which we feel may improve the algorithmic nature of these formulas.…”

Section: The Counting Formulassupporting

confidence: 61%

“…This is done in §7, it is just a reinterpretation of results of SzenesVergne [26] and it is connected to the combinatorial problem of understanding the regions (chambers) over which the partition function is a periodic polynomial. The answer is expressed by the notion of the Jeffrey-Kirwan residue.…”

Section: Theorem 52 the Subalgebra Of The Algebra Of Differential Fomentioning

confidence: 99%

“…The rest of this paper is devoted to show the relationship between our work and the theory of counting integer points in polytopes developed by Brion, Szenes, Vergne ( [4], [26]). In a way, it is mostly a reformulation in our language of their results and does not claim to be particularly original.…”

Section: Residuesmentioning

confidence: 99%

“…The knapsack problem. One of the aims of this paper is to understand in a more algebraic geometric language, the counting formulas of [5], [4], [26] for integral points in polytopes.…”

mentioning

confidence: 99%

“…In this paper we show that the counting formulas of [5], [4], [26], which are the higher dimensional case of the knapsack problem, can be interpreted in a similar way as sums of residues at the poles of the generating function. In order to develop this higher dimensional case, we shall study a trigonometric analog of the theory of hyperplane arrangements, which we call a toric arrangement, in which the main object of study is the open set of a torus complement of a finite set of cosets of codimension 1 tori.…”

mentioning

confidence: 99%

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Section: The Counting Formulassupporting

confidence: 61%

Section: Theorem 52 the Subalgebra Of The Algebra Of Differential Fomentioning

confidence: 99%

Section: Residuesmentioning

confidence: 99%

“…The knapsack problem. One of the aims of this paper is to understand in a more algebraic geometric language, the counting formulas of [5], [4], [26] for integral points in polytopes.…”

mentioning

confidence: 99%

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Wall-crossing formulas in Hamiltonian geometry

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2011

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NotationsThroughout the paper T will denote a compact, connected abelian Lie group, and t its Lie algebra. The integral lattice Λ ⊂ t is defined as the kernel of exp : t → T , and the real weight lattice Λ * ⊂ t * is defined by : Λ * := hom(Λ, 2πZ). Every µ ∈ Λ * defines a 1-dimensional T -representation, denoted by C µ , where t = exp X acts by t µ := e i µ,X . We denote by R(T ) the ring of characters of finite-dimensional Trepresentations. We denote by R −∞ (T ) the set of generalized characters of T . Anhas at most polynomial growth. The symplectic manifolds are oriented by their Liouville volume forms. If (Z, o Z ) is an oriented submanifold of an oriented manifold (M, o M ), we take on the fibers of the normal bundle N of Z in M , the orientation o N satisfying o M = o Z · o N . Duistermaat-Heckman measuresLet (M, Ω) be a symplectic manifold of dimension 2n equipped with an Hamiltonian action of a torus T , with Lie algebra t. The moment map Φ : M → t * satisfies the relations Ω(X M , −) + d Φ, X = 0, X ∈ t. We assume in this section that Φ is proper, and that the generic stabiliser Γ M of T on M is finite.The Duistermaat-Heckman measure DH(M) is defined as the pushforward by Φ of the Liouville volume form Ω n n! on M . For every f ∈ C ∞ (t * ) with compact support one has t * DH(M)(a)f (a) = M f (Φ) Ω n n! . In other terms DH(M)(a) = M δ(a − Φ) Ω n n! . We can define DH(M) in terms of equivariant forms as follows. Let A(M ) be the space of differential forms on M with complex coefficients. We denote by A −∞ temp (t, M ) the space of tempered generalized functions over t with values in A(M ), and by M −∞ temp (t * , M ) the space of tempered distributions over t * with values in A(M ). Let F : A −∞ temp (t, M ) → M −∞ temp (t * , M ) be the Fourier transform normalized by the condition that F (X → e i ξ,X ) is equal to the Dirac distribution a → δ(a − ξ).Let Ω t (X) = Ω − Φ, X be the equivariant symplectic form. We have then F (e −iΩt ) = e −iΩ δ(a − Φ) and so (2.1) DH(M) = (i) n M

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Residue formulae for vector partitions and Euler–MacLaurin sums

Cited by 74 publications

References 13 publications

On the geometry of toric arrangements

On the geometry of toric arrangements

Computing the Continuous Discretely

Wall-crossing formulas in Hamiltonian geometry

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